Use a check to determine whether the ordered triple is a solution of the system. See

Fill in the blanks. e x y  1 2x y  1 10 is called a ____ of equations.

Fill in the blanks. Because the ordered pair (2, 3) satisfies both equations in problem 1, it is a _____ of the system of equations.

Fill in the blanks. When the graphs of the equations of a system are identical lines, the equations are called dependent and the system has ______ many solutions.

Fill in the blanks. In the equation x 3y  1 x, the x-term has an understood ______ of 1.

Refer to the illustration. Determine whether a true or a false statement is obtained when the coordinates of a. point A are substituted into the equation for line l1 b. point are substituted into the equation for line l1 c. point C are substituted into the equation for line l1 d. point are substituted into the equation for line l2 y A B C l1 l2 111 1 2 3 4 2 3 4 2 2 3 4

Refer to the illustration. a. How many ordered pairs satisfy the equation 3x y  3? Name three. b. How many ordered pairs satisfy the equation 2 3 x y  3? Name three. c. How many ordered pairs satisfy both equations? Name it or them. y 3x + y = 3 x y = 3 2 3 111 1 2 3 4 5 6 7 2345 2 3

If the system e 4x 3y  7 3x 2y  6 is to be solved using the elimination method, by what constant should each equation be multiplied if a. the are to drop out? b. the are to drop out?

Consider the system: 2 3 x y 6  16 9 0.03x 0.02y  a. What step should be performed to clear the first equation of fractions? b. What step should be performed to clear the second equation of decimals?

Complete each solution. Solve: The solution is ( , 2). y  y  3( ) 7 y  3x 7 x  3 4x  x 7  5 x 3x 7  x ( )  5 e y  3x 7 x y  5 2 3 x

Complete each solution. Solve: The solution is ( , 3). y  2y  2y  7 x 2y  7 x  7x  x 2y  7 6x 2y  0 e 6x 2y  0 x 2y  7

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (4, 3); e ; 4x y  19 3x 2y  6 (4, 3

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (1, 2); e 3x y  5 x y  4 e (1

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (2, 3); y 2  1 2 x 3x 2y  0

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (1, 2); 2x y  0 y  1 2 x 3 2

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. a 1 2 , 1 3 b; 2x 3y  2 4x 9y  1 a

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. a3 4 , 2 3 e b 2; e 4x 3y  1 4x 3y  5 a3

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (0.2, 0.5); e 2x 5y  2.1 5x y  0.5

Use a check to determine whether the ordered pair is a solution of the system of equations. See Example 1. (2.1, 3.2); e x y  1.1 2x 3y  13.8 (2.

Solve each system by graphing. See Example 2. e x y  6 x y  2

Solve each system by graphing. See Example 2. e x y  4 2x y  5 e

Solve each system by graphing. See Example 2. e y  2x 1 x 2y  7

Solve each system by graphing. See Example 2. e 3x y  3 y  2x 7 e

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. e 3x 3y  4 x y  4

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. e 5x 2y  6 10x 4y  12 e

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. e x  3 2y 2x 4y  6

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. e 3x  5 2y 3x 2y  7

Solve each system by graphing. See Example 5. 1 6 x  1 3 y 1 2 y  x

Solve each system by graphing. See Example 5. x  y 3 1 4 x 1 6 y  1 3 1

Solve each system by graphing. See Example 5. 1 3 x 7 6 y  1 2 1 5 y  1 3 x 7 15 x

Solve each system by graphing. See Example 5. 3 5 x 1 4 y  11 10 1 8 x  13 24 1 3 y

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. e y  3.2x 1.5 y  2.7x 3.7

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. e y  0.45x 5 y  5.55x 13.7 e y

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. e 1.7x 2.3y  3.2 y  0.25x 8.95 e

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. e 2.75x  12.9y 3.79 7.1x y  35.76 e

Solve each system by substitution. See Example 6. e y  3x x y  8

Solve each system by substitution. See Example 6. e y  x 2 x 2y  16 e

Solve each system by substitution. See Example 6. e x  2 y 2x y  13

Solve each system by substitution. See Example 6. e x  5 y 3x 2y  7 e

Solve each system by substitution. See Example 6. e x 2y  6 3x y  10

Solve each system by substitution. See Example 6. e 2x y  21 4x 5y  7 e

Solve each system by substitution. See Example 6. e 5x 3y  26 3x y  14

Solve each system by substitution. See Example 6. e 3x 5y  4 5x y  14

Solve each system by elimination (addition). See Example 7. e x y  7 x y  11

Solve each system by elimination (addition). See Example 7. e a b  5 a b  11

Solve each system by elimination (addition). See Example 7. e 2s 3t  8 2s 3t  8

Solve each system by elimination (addition). See Example 7. e x 2y  21 x 2y  11 e

Solve each system by elimination (addition). See Example 7. e 3x 4y  24 5x 12y  72

Solve each system by elimination (addition). See Example 7. e 5x 2y  11 7x 6y  9

Solve each system by elimination (addition). See Example 7. 5 6 x 1 2 y  12 0.3x 0.5y  5.6

Solve each system by elimination (addition). See Example 7. 1 3 x 1 2 y  31 6 0.3x 0.2y  3.9

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 8 and 9. e 6x 3y  18 y  2x 5

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 8 and 9. e 8x 4y  16 2x 4  y

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 8 and 9. e 3x y  5 21x  7(y 5)

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 8 and 9. e 4x 8y  15 x  2(2 y)

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) e 4x 3y  5 y  2x

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) e 2x 2y  1 3x 4y  0 e 4

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) y  5 2 x 1 2 2x 3 2 y  5

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) 5 2 x 3y  6 y  5 6 x 2

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) x  11 2y 3 y  11 6x 4

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) x  1 3y 4 y  12 3x 2

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) e x  13 4y 3x  4 2y

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) e 3x  7 2y 2x  2 4y e

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) x  2 y  1 2 x 2 e

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) y  2 y  2 3 x 4 3 x

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) x 3y  6 y  1 3 x 2 y

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) 2x y  4 2y  4x 6 x

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. e 2x 3y  8 3x 2y  1

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. x  3 2 y 5 2x 3y  8

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. e 4(x 2)  9y 2(x 3y)  3 x 

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. e 2(2x 3y)  5 8x  3(1 3y)

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 0.3a 0.1b  0.5 4 3 a 1 3 b  3 e

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 0.9p 0.2q  1.2 2 3 p 1 9 q  1 0

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. x 2 y 2  6 x 3 y 3  4 0

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. x 2 y 3  4 x 2 y 9  0

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. x  2 3 y y  4x 5

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 5x 2y  19 y  1 3x 4 x

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. e 3x 4y  9 x 2y  8

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. e 3x 2y  10 6x 5y  25 e

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. x 4y 5  4 y 3  x 2 5 2 e

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 3x 2y  9 2 x 2 3 4  2y

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 2 3 x 1 4 y  8 0.5x 0.375y  9

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 0.5x 0.5y  6 x 2 y 2  2 2

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. 3 2 p 1 3 q  2 2 3 p 1 9 q  1 0.

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. a b 3  5 3 a b 3  3 a

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. m n 5 m n 2  6 m n 2 m n 4  3

Solve each system by any method. If a system is inconsistent or if the equations are dependent, so indicate. r 2 5 s 3 2  5 r 3 2 s 2 3  6

Solve each system. To do so, substitute a for 1 x and b for 1 y and solve for a and b. Then find x and y using the fact that a 1 x and b 1 y a 1 x 1 y  5 6 1 x 1 y  1 6 b

Solve each system. To do so, substitute a for 1 x and b for 1 y and solve for a and b. Then find x and y using the fact that a 1 x and b 1 y a 1 x 1 y  9 20 1 x 1 y  1 20

Solve each system. To do so, substitute a for 1 x and b for 1 y and solve for a and b. Then find x and y using the fact that a 1 x and b 1 y a 1 x 2 y  1 2 x 1 y  7

Solve each system. To do so, substitute a for 1 x and b for 1 y and solve for a and b. Then find x and y using the fact that a 1 x and b 1 y a 3 x 2 y  30 2 x 3 y  30

Fashion Designer One of the line graphs below gives the percent share of the U.S. footwear market for shoes produced in the United States. The other line gives the percent share of the U.S. footwear market for imports. a. Estimate the coordinates of the point of intersection of the graphs. b. What important percent-of-the-market information does your answer to part a give? 100% 80% 60% 40% 20% 0% 1960 1970 1980 Year Source: Americal Apparel and Footwear Association 1990 2000 2010 Produced in U.S. Imports Share of U.S. Footware Market

The Internet. The graph on the next page shows the growing importance of the Internet in the daily lives of Americans. Determine when the time spent on the following activities was the same. Approximately how many hours per year were spent on each? 250 Hours per year per person 99 00 01 02 03 04 05 06 07 08 09 Year Source: Vernois Suhler Stevenson Consumer Internet Daily newpapers Consumer magazines

Law of Supply and Demand. The demand function, graphed below, describes the relationship between the price of a certain camera and the demand for the camera. a. The supply function, S(x)  25 4 x 525, describes the relationship between the price of the camera and the number of cameras the manufacturer is willing to supply. Graph this function in the illustration. b. For what price will the supply of cameras equal the demand? c. As the price of the camera is increased, what happens to supply and what happens to demand? y x 100 100 200 300 400 500 600 120 140 Demand function Price per camera ($) Number of cameras 160 180 200

Cost and Revenue. The function C(x)  200x 400 gives the cost for a college to offer sections of an introductory class in CPR (cardiopulmonary resuscitation). The function R(x)  280x gives the amount of revenue the college brings in when offering sections of CPR. a. Find the break-even point (where cost revenue) by graphing each function on the same coordinate system. b. How many sections does the college need to offer to make a profit on the CPR training course?

Business. Estimate the break-even point (where cost revenue) on the graph in the next column. Explain why it is called the break-even point. 1,000 2,000 3,000 4,000 $5,000 100 200 Number of widgets produced Total revenue Total cost 300 400 500 Production costs and

Navigation. The paths of two ships are tracked on the same coordinate system. One ship is following a path described by the equation 2x 3y  6, and the other is following a path described by the equation y  2 3 x 3. a. Is there a possibility of a collision? b. What are the coordinates of the danger point? c. Is a collision a certainty?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Ticket Sales. The ticket prices for a Halloween haunted house were $5 for adults and $3 for children. On a day when a total of 390 tickets were purchased, the receipts were $1,470. How many of each type of ticket were sold?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Advertising. Use the information in the ad to find the cost of a 15-second and the cost of a 30-second radio commercial on radio station KLIZ. ADVERTISE YOUR COMPANY ON THE RADIO KLIZ 1250 AM Plan 1: Four 30-second spots, six 15-second spots Cost: $6,050 Plan 2: Three 30-second spots, five 15-second spots Cost: $4,775

In Exercises 97112, write a system of two equations in two variables to solve each problem. Fashion Designer In 2009, there was a combined total of 4,046 Gap and Aropostale clothing stores worldwide. The number of Gap stores was 31 4 times more than the number of Aropostale stores. How many Gap stores and how many Aropostale stores were there that year? (Source: wikinvest.com)

In Exercises 97112, write a system of two equations in two variables to solve each problem. Summer Concerts. According to StubHub.com, in 2009, two tickets to a Jonas Brothers concert and two tickets to an Elton John concert cost, on average, a total of $560. At those prices, four tickets to see the Jonas Brothers and two tickets to see Elton John cost $806. What was the average cost of a Jonas Brothers ticket and an Elton John ticket in 2009?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Geometry. An acute angle is an angle with measure less than 90 In a right triangle, the measure of one acute angle is 15 greater than two times the measure of the other acute angle. Find the measure of each acute angle.

In Exercises 97112, write a system of two equations in two variables to solve each problem. New York City. The triangular-shaped Flatiron Building in Manhattan has a perimeter of 499 feet at its base. It is bordered on each side by a street. The 5th Avenue front of the building is 198 feet long. The Broadway front is 43 feet more than twice as long as the East 22nd Street front. Find the length of the Broadway front and East 22nd Street front. (Source: New York Public Library)

In Exercises 97112, write a system of two equations in two variables to solve each problem. Investment Clubs. Part of $8,000 was invested by an investment club at interest and the rest at If the annual income from these investments is how much was invested at each rate?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Retirement Income. A retired couple invested part of $12,000 at interest and the rest at If their annual income from these investments is $810, how much was invested at each rate?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Snowmobiling. A man rode a snowmobile at the rate of 20 mph and then skied cross country at the rate of 4 mph. During the 6-hour trip, he traveled 48 miles. How long did he snowmobile, and how long did he ski?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Salmon. It takes a salmon 40 minutes to swim 10,000 feet upstream and 8 minutes to swim that same portion of a river downstream. Find the speed of the salmon in still water and the speed of the current.

In Exercises 97112, write a system of two equations in two variables to solve each problem. Production Planning. A manufacturer builds racing bikes and mountain bikes, with the per unit manufacturing costs shown in the table. The company has budgeted $26,150 for materials and $31,800 for labor. How many bicycles of each type can be built? Model Cost of materials Cost of labor Racing $110 $120 Mountain $140 $180

In Exercises 97112, write a system of two equations in two variables to solve each problem. Farming. A farmer keeps some animals on a strict diet. Each animal is to receive 15 grams of protein and 7.5 grams of carbohydrates. The farmer uses two food mixes, with nutrients as shown in the table. How many grams of each mix should be used to provide the correct nutrients for each animal? Mix Protein Carbohydrates Mix A 12% 9% Mix B 15% 5%

In Exercises 97112, write a system of two equations in two variables to solve each problem. Cosmotology. A beauty shop specializing in permanents has fixed costs of $2,101.20 per month. The owner estimates that the cost for each permanent is $23.60, which covers labor, chemicals, and electricity. If her shop can give as many permanents as she wants at a price of $44 each, how many must be given each month for her to break even?

In Exercises 97112, write a system of two equations in two variables to solve each problem. Mixing Candy. How many pounds of each candy shown in the illustration must be mixed to obtain 60 pounds of candy that would be worth $4 per pound? Gummy Bears $3.50/lb Jelly Beans $5.50/lb

In Exercises 97112, write a system of two equations in two variables to solve each problem. Dermatology. Tests of an antibacterial face-wash cream showed that a mixture containing 0.3% Triclosan (active ingredient) gave the best results. How many grams of cream from each tube should be used to make an equal-size tube of the 0.3% cream? Contents: 185 g Daily Face Wash 0.2% Triclosan Contents: 185 g Daily Face Wash 0.7% Triclosan

In Exercises 97112, write a system of two equations in two variables to solve each problem. Mixing Solutions. How many ounces of the two alcohol solutions in the illustration must be mixed to obtain 100 ounces of a 12.2% solution? 8% 15% 100 oz 12.2%

Which method would you use to solve the system? Explain. e y 1  3x 3x 2y  12

Which method would you use to solve the system? Explain. e 2x 4y  9 3x 5y  20

When solving a system, what advantages are there with the substitution and elimination methods compared with the graphing method?

When using the elimination (addition) method, how can you tell whether a. a system of linear equations has no solution? b. a system of linear equations has infinitely many solutions?

Solve each formula for the specified variable. V2 V1  P1 P2 for P1

Solve each formula for the specified variable. 1 r  1 r1 1 r2 for r

Solve each formula for the specified variable. S  r a lr 1 r for r

Solve each formula for the specified variable. P  Q1 Q1 Q2 for Q1

If the solution of the system Ax By  2 Bx Ay  26 P  is (3, 5), find the values of A and B.

Solve and assume that and are constants. e db 2ab 3cd  1 3ab 2cd  1 A

Fill in the blanks. 2x y 3z  0 3x y 4z  5 4x 2y 6z  0 is called a ______ of three linear equations in three variables. Each equation is written in ______Ax By Cz  Dform.

Fill in the blanks. If the first two equations of the system in Exercise 1 are added, the variable y is ______ .

Fill in the blanks. Solutions of a system of three equations in three variables, x, y, and z, are written in the form (x, y, z) and are called ordered ______.

Fill in the blanks. The graph of the equation 2x 3y 4z  5 z is a flat surface called a _____ .

Fill in the blanks. When three planes coincide, the equations of the system are ______ , and there are infinitely many solutions.

Fill in the blanks. When three planes intersect in a line, the system will have _____ many solutions.

For each graph of a system of three equations, determine whether the solution set contains one solution, infinitely many solutions, or no solution. a. b.

Consider the system: 2x y 4z  3 x y 2z  1 x y 3z  2 (1) (2) a. What is the result if equation 1 and equation 2 are added? b. What is the result if equation 2 and equation 3 are added? c. What variable was eliminated in the steps performed in parts (a) and (b)?

For the following system, clear the equations of any fractions or decimals and write each equation in Ax By Cz  D S form. x y  3 4z 0.7x 0.2y 0.8z  1.5 x 2 y 3 z 6  2 3 Ax B

What is the purpose of the numbers shown in red in front of the equations below? (1) (2) (3) x y z  6 2x y z  3 5x 3y z  2 (1)

Use a check to determine whether the ordered triple is a solution of the system. See Example 1. (2, 1, 1) x y z  2 2x y z  4 2x 3y z  2 (2, 1

Use a check to determine whether the ordered triple is a solution of the system. See Example 1. (3, 2, 1) 3x y z  6 2x 2y 3z  1 x y 2z  1 x y

Use a check to determine whether the ordered triple is a solution of the system. See Example 1. (6, 7, 5) 3x 2y z  37 x 3y  27 2x 7y 2z  48

Use a check to determine whether the ordered triple is a solution of the system. See Example 1. (4, 0, 9) x 2y 3z  31 2x 6z  46 3x y  12

Solve each system. See Example 2. x y z  4 2x y z  1 2x 3y z  1

Solve each system. See Example 2. x y z  4 x y z  2 x y 2z  1

Solve each system. See Example 2. 3x 2y 5z  3 4x 2y 3z  10 5x 2y 2z  11

Solve each system. See Example 2. 5x 4y 2z  2 3x 4y 3z  27 2x 4y 7z  23

Solve each system. See Example 2. 2x 6y 3z  9 5x 3y 5z  3 4x 3y 2z  15

Solve each system. See Example 2. 4x 3y 5z  23 2x 5y 3z  13 4x 6y 7z  7

Solve each system. See Example 2. 4x 5y 8z  52 2x 3y 4z  26 3x 7y 8z  31

Solve each system. See Example 2. 2x 6y 3z  20 5x 3y 5z  47 4x 3y 2z  4

Solve each system. See Example 3. 3x 3z  6 4y 7x 5z  46 2y 4x  31 z

Solve each system. See Example 3. 5x 6z  4y 21 9x 2y  3z 47 3x y  19

Solve each system. See Example 3. 2x z  2 y 8x 3y  2 6x 2y 3z  4

Solve each system. See Example 3. 3y z  1 x 2z  9 6y 9y 3z  9 2x

Solve each system using elimination. See Example 4. x y 3z  35 x 3y  20 2y z  35

Solve each system using elimination. See Example 4. x 2y 3z  11 5x y  13 2x 3z  11

Solve each system using elimination. See Example 4. 3x 2y z  27 6x 3y  2 3y 2z  8

Solve each system using elimination. See Example 4. 2x y 4 x 2y  8z 7 y 4z  5

Solve each system using substitution. See Example 5. r s 3t  21 r 4s  9 5s t  4

Solve each system using substitution. See Example 5. r s 6t  12 r 6s  28 7s t  26

Solve each system using substitution. See Example 5. x 8z  30 3x y 4z  5 y 7z  30

Solve each system using substitution. See Example 5. x 6z  36 5x 3y 2z  20 y 4z  20

Solve each system. If a system is inconsistent or if the equations are dependent, state this. See Examples 6 and 7. 7a 9b 2c  5 5a 14b c  11 2a 5b c  3

Solve each system. If a system is inconsistent or if the equations are dependent, state this. See Examples 6 and 7. 3x 4y z  10 x 2y z  3 2x y z  5

Solve each system. If a system is inconsistent or if the equations are dependent, state this. See Examples 6 and 7. 7x y z  10 x 3y z  2 x 2y z  1

Solve each system. If a system is inconsistent or if the equations are dependent, state this. See Examples 6 and 7. 2a b c  6 5a 2b 4c  30 a b c  8

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2a 3b 2c  18 5a 6b c  21 4b 2c 6  0

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. r s t  4 r 2s t  1 r s 3t  2

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x 2y z  2 x 3z 24  0 y  7 4z

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. r 3t  11 r s t  13 s 4t  12

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. b 2c  7 a a c  2(4 b) 2a b c  9

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 0.02a  0.02 0.03b 0.01c 4a 6b 2c 5  0 a c  3 2b

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x y z  1 x 2y 2z  2 4x 5y 3z  3

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x 2y 3z  10 3x y z  0 x y 2z  6

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 0.4x 0.3z  0.4 2y 6z  1 4(2x y)  9 3z

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. a b c  180 a 4 b 2 c 3  60 2b 3c 330  0 0

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. r s 4t  3 3r 7t  0 3s 5t  0

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x y  3 2x y z  1 x z  2

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 0.5a 0.3b  2.2 1.2c 8.5b  24.4 3.3c 1.3a  29

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 4a 3b  1 6a 8c  1 2b 4c  0

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x 3y  6 4z 2x  3y 4z 4 4x 6y 8z  12

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x 5y 7z  0 4x y z  0 x y 4z  0 2x

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. a b  2 c a  3 b c a b c 4  0

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 0.1x 0.3y 0.4z  0.2 2x y 2z  3 4x 5y 10z  7

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x 1 3 y z  13 1 2 x y 1 3 z  2 x 1 2 y 1 3 z  2

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x 1 5 y z  9 1 4 x 1 5 y 1 2 z  5 2x y 1 6 z  12

Graphs of Systems. Explain how each of the following pictures is an example of the graph of a system of three equations. Then describe the solution, if there is any. a. b. c. d

Zoology. An X-ray of a mouse revealed a cancerous tumor located at the intersection of the coronal, sagittal, and transverse planes. From this description, would you expect the tumor to be at the base of the tail, on the back, in the stomach, on the tip of the right ear, or in the mouth of the mouse? Transverse plane Coronal plane

NBA Records. The three highest one-game point totals by one player in a National Basketball Association game are shown below. Solve the following system to find , , and . x y z  259 x y  19 x z  22 zyx Sagittal plane Transverse plane Coronal plane 12.2 Solving Systems of Equations in Three Variables 1029 Pts Player, team Date Wilt Chamberlain, Philadelphia 3/2/1962 Kobe Bryant, Los Angeles 1/22/2006 z Wilt Chamberlain, Philadelphia 12/8/1961 y

Bicycle Frames. The angle measures of the triangular part of the bicycle frame shown can be found by solving the following system. Find , , and . x y z  180 x y  120 y z  135 zyx z x y Seat tube Handle bar tube WRI

Explain how a system of three equations in three variables can be reduced to a system of two equations in two variables.

What makes a system of three equations with three variables inconsistent?

What does the graph of a linear equation in three variables such as 2x 3y 9z = 10 look like?

What situation discussed in this section looks like two walls of a room and the floor meeting in a corner?

Graph each of the basic functions. (x) x

Graph each of the basic functions. g(x)  x2

Graph each of the basic functions. h(x)  x 3

Graph each of the basic functions. S(x)  x

Solve each system. w x y z  3 w x y z  1 w x y z  1 w x y z  3

Solve each system. 1 x 1 y 1 z  3 2 x 1 y 1 z  0 1 x 2 y 4 z  21

Solve each system. 4a b 2c 3d  16 3a 3b c 4d  20 a 2b 5c d  4 5a 4b 3c d  10

Solve each system. a c 2d  4 b 2c  1 a 2b c  2 2a b 3c 2d  4

Fill in the blanks. If a point lies on the graph of an equation, it is a solution of the equation, and the coordinates of the point _____ the equation.

Fill in the blanks. The process of determining an equation whose graph contains given points is called curve _____ .

Write a system of three equations in three variables that models the situation. Do not solve the system. 3. Desserts. A bakery makes three kinds of pies: chocolate cream, which sells for $5; apple, which sells for $6; and cherry, which sells for $7. The cost to make the pies is $2, $3, and $4, respectively. Let the number of chocolate cream pies made daily, the number of apple pies made daily, and the number of cherry pies made daily. Each day, the bakery makes 50 pies. Each day, the revenue from the sale of the pies is $295. Each day, the cost to make the pies is $145.

Write a system of three equations in three variables that models the situation. Do not solve the system. Fast Foods. Let the number of calories in a Big Mac hamburger, the number of calories in a small order of French fries, and the number of calories in a medium Coca-Cola. The total number of calories in a Big Mac hamburger, a small order of French fries, and a medium Coke is 1,000. The number of calories in a Big Mac is 260 more than in a small order of French fries. The number of calories in a small order of French fries is 40 more than in a medium Coke. (Source: McDonalds USA)

What equation results when the coordinates of the point (2, 3) are substituted into y  ax2 bx c?

The equation y  5x2 6x 1 is written in the form y  ax2 bx c. What are a, b, and c?

Making Statues. An artist makes three types of ceramic statues (large, medium, and small) at a monthly cost of $650 for 180 statues. The manufacturing costs for the three types are $5, $4, and $3. If the statues sell for $20, $12, and $9, respectively, how many of each type should be made to produce $2,100 in monthly revenue?

Puppets. A toy company makes a total of 500 puppets in three sizes during a production run. The small puppets cost $5 to make and sell for $8 each, the standard-size puppets cost $10 to make and sell for $16 each, and the super-size puppets cost $15 to make and sell for $25. The total cost to make the puppets is $4,750 and the revenue from their sale is $7,700. How many small, standard, and super-size puppets are made during a production run?

Nutrition. A dietician is to design a meal using Foods A, B, and C that will provide a patient with exactly 14 grams of fat, 13 grams of carbohydrates, and 9 grams of protein. Each ounce of Food A contains 2 grams of fat, 3 grams of carbohydrates, and 2 grams of protein. Each ounce of Food B contains 3 grams of fat, 2 grams of carbohydrates, and 1 gram of protein. Each ounce of Food C contains 1 gram of fat, 1 gram of carbohydrates, and 2 grams of protein. a. Complete the following table and then form a system of three equations that could be used to determine how many ounces of each food should be used in the meal. Name Number of Grams of Grams of Grams of of food ounces used fat carbohydrates protein A a 2a 3a B b 3b b C c c 2c Total: 14 Total: Total: 9 b. Solve the system from part a.

Nutritional Planning. One ounce of each of three foods has the vitamin and mineral content shown in the table. How many ounces of each must be used to provide exactly 22 milligrams (mg) of niacin, 12 mg of zinc, and 20 mg of vitamin C? Milligrams per ounce in each food type Food Niacin Zinc Vitamin C A 1 mg 1 mg 2 mg B 2 mg 1 mg 1 mg C 2 mg 1 mg 2 mg

Fashion Designer A clothing manufacturer makes coats, shirts, and slacks. The time required for cutting, sewing, and packaging each item is shown in the table. How many of each should be made to use all available labor hours? Coats Shirts Slacks Time available Cutting 20 min 15 min 10 min 115 hr Sewing 60 min 30 min 24 min 280 hr Packaging 5 min 12 min 6 min 65 hr

Sculpting. A wood sculptor carves three types of statues with a chainsaw. The number of hours required for carving, sanding, and painting a totem pole, a bear, and a deer are shown in the table. How many of each should be produced to use all available labor hours? Totem pole Bear Deer Time available Carving 2 hr 2 hr 1 hr 14 hr Sanding 1 hr 2 hr 2 hr 15 hr Painting 3 hr 2 hr 2 hr 21 hr

NFL Records. Jerry Rice, who played the majority of his career with the San Francisco 49ers and the Oakland Raiders, holds the all-time record for touchdown (TD) passes caught. Here are some interesting facts about this feat. He caught 30 more TD passes from Steve Young than he did from Joe Montana. He caught 39 more TD passes from Joe Montana than he did from Rich Gannon. He caught a total of 156 TD passes from Young, Montana, and Gannon. Determine the number of touchdown passes Rice has caught from Young, from Montana, and from Gannon.

Hot Dogs. In 10 minutes, the top three finishers in the 2010 Nathans Hot Dog Eating Contest consumed a total of 136 hot dogs. The winner, Joey Chestnut, ate 9 more hot dogs than the runner-up, Tim Janus. Pat Bertoletti finished a distant third, 8 hot dogs behind Janus. How many hot dogs did each person eat? (Source: nathansfamous.com)

Earths Atmosphere. Use the information in the circle graph to determine what percent of Earths atmosphere is nitrogen, is oxygen, and is other gases. Nitrogen: This is 12% more than three times the sum of the percent oxygen and the percent other gases. Other gases: This is 20% less than the percent oxygen. Nitrogen Oxygen

Deceased Celebrities. Between October 2009 and October 2010, the estates of Michael Jackson, Elvis Presley, and J.R.R. Tolkien (author of Lord of the Rings) earned a total of $385 million. Together, the Presley and Tolkien estates earned $165 million less than the Jackson estate. The Jackson estate earned 5.5 times as much as the Tolkien estate. Use this information to label each bar on the graph below. (Source: Forbes.com) Michael Jackson (19582009) Elvis Presley (19351977) J.R.R. Tolkein (18921973) The Three Top-Earning Deceased Celebrities (20092010) Millions of dollars

Triangles. The sum of the measures of the angles of any triangle is . In DABC, A measures 100 less than the sum of the measures of B and C, and the measure of C is 40 less than twice the measure of B. Find the measure of each angle of the triangle.

Quadrilaterals. A quadrilateral is a four-sided polygon. The sum of the measures of the angles of any quadrilateral is 360. In the illustration, the measures of A and B are the same. The measure of A is 20 greater than the measure of A, and the measure of D is 60 less than B. Find the measure of , B, , and . C

TV History. X-Files, Will & Grace, and Seinfeld are three of the most popular television shows of all time. The total number of episodes of these three shows is 575. There are 21 more episodes of X-Files than Seinfeld, and the difference between the number of episodes of Will & Grace and Seinfeld is 14. Find the number of episodes of each show.

Traffic Lights. At a traffic light, one cycle through green-yellowred lasts for 80 seconds. The green light is on eight times longer than the yellow light, and the red light is on eleven times longer than the yellow light. For how long is each colored light on during one cycle?

Ice Skating. Three circles are traced out by a figure skater during her performance as shown below. If the centers of the circles are the given distances apart (10 yd, 14 yd, and 18 yd), find the radius of each circle. (Hint: Each red line segment is composed of two radii. Label the radii r1, r2, and r3). 10 yd 14 yd Circle 1 Circle 2 C

NBA Centers. Together, Shaquille ONeil (Boston Celtics) and Dwight Howard (Orlando Magic) weigh 590 pounds. Together ONeil and Yao Ming (Houston Rockets) weigh 635 pounds. Together, Howard and Yao Ming weigh 575 pounds. Find the weight of each center.

Potpourri. The owner of a home decorating shop wants to mix dried rose petals selling for $6 per pound, dried lavender selling for $5 per pound, and buckwheat hulls selling for $4 per pound to get 10 pounds of a mixture that would sell for $5.50 per pound. She wants to use twice as many pounds of rose petals as lavender. How many pounds of each should she use?

Mixing Nuts. The owner of a candy store wants to mix some peanuts worth $3 per pound, some cashews worth $9 per pound, and some Brazil nuts worth $9 per pound to get 50 pounds of a mixture that will sell for $6 per pound. She uses 15 fewer pounds of cashews than peanuts. How many pounds of each did she use?

Piggy Banks. When a child breaks open her piggy bank, she finds a total of 64 coins, consisting of nickels, dimes, and quarters. The total value of the coins is $6. If the nickels were dimes, and the dimes were nickels, the value of the coins would be $5. How many nickels, dimes, and quarters were in the piggy bank?

Theater Seating. The illustration shows the cash receipts and the ticket prices from two sold-out Sunday performances of a play. Find the number of seats in each of the three sections of the 800-seat theater. Box seat tickets Matinee $20 Evening $30 Founder's circle tickets Matinee $30 Evening $40 Stage Promenade tickets Matinee $10 Evening $25 Sunday Ticket Receipts Matinee $13,000 Evening $23,000

Astronomy. Comets have elliptical orbits, but the orbits of some comets are so large that they look much like a parabola. Find an equation of the form y  ax2 bx c for the parabola that closely describes the orbit of the comet shown in the illustration. 2, 5) y x Sun 2 4 6 2

Curve Fitting. Find an equation of the form y  ax2 bx c for the parabola shown in the illustration. (1, 1) (3, 7) (1, 3) x y

Walkways. A circular sidewalk is to be constructed in a city park. The walk is to pass by three particular areas of the park, as shown in the illustration. If an equation of a circle is of the form x 2 y2 Cx Dy E  0, find an equation that describes the path of the sidewalk by determining , , and . x y City Park Wishing well Fish pond Circular walkway Rose garden (3, 1) (1, 3) (1, 1) 1 4 1 5 2 2345

Curve Fitting. The equation of a circle is of the form x2 y2 Cx Dy E  0. Find an equation of the circle shown in the illustration by determining C, D, and E. 6, 0) (3, 3) (0, 0) x y 1 1 1 2 3 4

Explain why the following problem does not give enough information to answer the question: The sum of three integers is 48. If the first integer is doubled, the sum is 60. Find the integers.

Write an application problem that can be solved using a system of three equations in three variables.

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. y   1 x

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. y 4   x

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. xy  9

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. y  0 x

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. x 1  0 y 0

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. y  1 x2

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. y2  x

Determine whether each equation defines to be a function of . If it does not, find two ordered pairs where more than one value of corresponds to a single value of x. x  0 y 0 2

Digits Problems. The sum of the digits of a three-digit number is 8. Twice the hundreds digit plus the tens digit is equal to the ones digit. If the digits of the number are reversed, the new number is 82 more than twice the original number. What is the three-digit number?

Purchasing Pets. A pet store owner spent $100 to buy 100 animals. He bought at least one iguana, one guinea pig, and one mouse, but no other kinds of animals. If an iguana cost $10.00, a guinea pig cost $3.00, and a mouse cost $0.50, how many of each did he buy?

Fill in the blanks. A ______ is a rectangular array of numbers written within brackets.

Fill in the blanks. Each number in a matrix is called an _____ or entry of the matrix.

Fill in the blanks. If the order of a matrix is 3  4, it has 3 _____ and 4 _____ . We read 3  4 as 3 _____ 4.

Fill in the blanks. A matrix that represents the equations of a system is called an ______ matrix.

Fill in the blanks. Elementary ______ operations can be used on an augmented matrix to produce a simpler equivalent matrix that gives the solution of a system. This process is called ______ - ______. elimination.

Fill in the blanks. The matrix 1 0 2 01 4 d, with 1s down its main ______ and 0s directly above and below it, is in reduced row- ______ form.

For each matrix, determine the number of rows and the number of columns. a. c 4 6 1 1 9 3 d c b. 1 231 0 164 0 01 1 3

Fill in the blanks to complete each elementary row operation: a. Type 1: Any two rows of a matrix can be _____ . b. Type 2: Any row of a matrix can be ____ by a nonzero constant. c. Type 3: Any row of a matrix can be changed by _____ a nonzero constant multiple of another row to it.

Gauss-Jordan elimination was used to solve a system. The final augmented matrix is shown. Fill in the blanks. a. c 1 0 10 01 6 d represents e 10 6 c The solution of the system is ( , ). b. 100 16 010 8 001 4 represents 16 y 8 4 The solution of the system is ( , , ).

a. Which matrix shown below indicates that its corresponding system of equations has no solution? b. Which matrix indicates that the equations of its corresponding system are dependent? i. c d 1 2 4 00 0 d ii. c d 106 010 c d 1 iii. c 1 2 4 00 2 c d 10

Explain what each symbolism means. a. R1 4 R2 b. 1 2 R1 c. 6R2  R3

Complete the solution to solve the system using matrices. e 4x y 14 x  y 6 104 012 d  1 5 R2 c 116 0 1 2 d 4R1 c 1 1 6 0 10 d R1 4 c 1 6 4 1 14 d c 4 This augmented matrix represents the equivalent system x 4 2  The solution is ( , 2) .

Represent each system using an augmented matrix. See Example 1. e x  2y 6 3x y 10

Represent each system using an augmented matrix. See Example 1. x  y  z 4 2x  y z 1 2x 3y 1

For each augmented matrix, give the system of equations that it represents. c 167 014 d

For each augmented matrix, give the system of equations that it represents. 1 29 1 0 14 0 0 01 7

Perform each of the following elementary row operations on the augumented matrix c 3 1 6 1 4 4 d. See Example 2. R1 4 R2

Perform each of the following elementary row operations on the augumented matrix c 3 1 6 1 4 4 d. See Example 2. 5R2

Perform each of the following elementary row operations on the augumented matrix c 3 1 6 1 4 4 d. See Example 2. 1 3 R1

Perform each of the following elementary row operations on the augumented matrix c 3 1 6 1 4 4 d. See Example 2. 3R2  R1

Perform each of the following elementary row operations on the augmented matrix 3 6 9 0 1 5 2 1 2 2 2 5 . See Example 2. R2 4 R3

Perform each of the following elementary row operations on the augmented matrix 3 6 9 0 1 5 2 1 2 2 2 5 . See Example 2. 1 2R3

Perform each of the following elementary row operations on the augmented matrix 3 6 9 0 1 5 2 1 2 2 2 5 . See Example 2. R1  R2

Perform each of the following elementary row operations on the augmented matrix 3 6 9 0 1 5 2 1 2 2 2 5 . See Example 2. 2R2  R3

Use matrices to solve each system of equations. See Example 3. e x  y 2 x y 0

Use matrices to solve each system of equations. See Example 3. e x  y 3 x y 1 e x

Use matrices to solve each system of equations. See Example 3. e 2x  y 1 x  2y 4

Use matrices to solve each system of equations. See Example 3. e 5x 4y 10 x 7y 2 e

Use matrices to solve each system of equations. See Example 4. x  y  z 6 x  2y  z 8 x  y  2z 7

Use matrices to solve each system of equations. See Example 4. x  y  z 6 x  2y  z 8 x  y  2z 9

Use matrices to solve each system of equations. See Example 4. 3x  y 3z 5 x 2y  4z 10 x  y  z 13

Use matrices to solve each system of equations. See Example 4. 2x  y 3z 1 3x 2y z 5 x 3y 2z 12 3x  y

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. See Example 5. e x 3y 9 2x  6y 18

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. See Example 5. e 6x  12y 10 2x 4y 8 e

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. See Example 5. e 4x 4y 12 x  y 3 e 6

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. See Example 5. e 5x 15y 10 x 3y 2 e

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2x  3y z 8 x y z 2 4x  3y  z 6

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2a  b  3c 3 2a b  c 5 4a 2b  2c 2

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 2x y 1 x 2y 1

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 2x y 0 x  y 3 e

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 3x  4y 12 9x 2y 6

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 2x 3y 16 4x  y 22

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2x  y z 1 x  2y  2z 2 4x  5y  3z 3

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. x y 1 2x z 0 2y z 2

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 8x 2y 4 4x y 2

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. e 9x 3y 6 3x y 8

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2x  y 2z 6 4x y  z 1 6x 2y  3z 5

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2x 3y  3z 14 3x  3y z 2 2x  6y  5z 9

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 6x  y z 2 x  2y  z 5 5y z 2

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 2x  3y 2z 18 5x 6y  z 21 4y 2z 6

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. 5x  3y 4 3y 4z 4 x  z 1

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. y  2z 2 x  y 1 2x z 0

Digital Photography. A digital camera stores the black and white photograph shown on the right as a 512  512 matrix. Each element of the matrix corresponds to a small dot of grey scale shading, called a pixel, in the picture. How many elements does a 512  512 matrix have? 512 400 300 200 100

Digital Imaging. A scanner stores a black and white photograph as a matrix that has a total of 307,200 elements. If the matrix has 480 rows, how many columns does it have?

Complementary Angles. The following system can be used to find the angle measures (in degrees) of the two complementary angles shown in the illustration below. Solve the system using matrices to find the measure of each angle. (Hint: Each equation must be in standard Ax By C form.)

Supplementary Angles. The following system can be used to find the angle measures (in degrees) of the two supplementary angles shown in the illustration below. Solve the system using matrices to find the measure of each angle. (Hint: Each equation must be in standard Ax  By C form.) e x  y 180 y x 14 Ax y x

Triangles. The following system can be used to find the measures (in degrees) of and shown in the illustration below. Solve the system using matrices to find the measure of each angle of the triangle. (Hint: Each equation must be in standard Ax + By = C form.) A  B  C 180 B A  25 C 2A 5 A, A B C

Triangles. The following system can be used to find the measures (in degrees) of and shown in the illustration below. Solve the system using matrices to find the measure of each angle of the triangle. (Hint: Each equation must be in standard Ax + By = C form.) A  B  C 180 A B 10 B C 10 A, B A B C

For the system e 2x 3y 5 4x  8 y 104, explain what is wrong with writing its corresponding augmented matrix as c 2 3 5 4 81 d e. How should it be written?

Explain what is meant by the phrase reduced row-echelon form. Give an example of how it was used in this section.

Explain how a type 3 row operation is similar to the elimination (addition) method of solving a system of equations.

If the system represented by the augmented matrix at the right has no solution, what do you know about ? Explain your answer. 1101 0012 000 k

What is the formula used to find the slope of a line, given two points on the line?

What is the form of the equation of a horizontal line? Of a vertical line?

What is the pointslope form of the equation of a line?

What is the slopeintercept form of the equation of a line?

Use matrices to solve the system. x2  y2  z 2 14 2x2  3y2 2z 2 7 x2 5y2  z 2 8 A B C

Use matrices to solve the system. w  x  y  z 0 w 2x  y 3z 3 2w  3x  y 2z 1 2w 2x 2y  z 12 x2  y2

Fill in the blanks. ` 4 9 6 1 `is a _____ . The numbers 4 and 1 lie along its main _____.

Fill in the blanks. A determinant is number that is associated with a _____ matrix.

Fill in the blanks. The _____ of in a1 b1 c1 a2 b2 c2 a3 b3 c3 is ` a2 c2 a3 c3 `

Fill in the blanks. ______ rule uses determinants to solve systems of linear equations.

Fill in the blanks. ` a b c d `

Fill in the blanks. To find the minor of 5, we cross out the elements of the determinant that are in the same row and column as . 3 51 6 2 2 8 1 4

Fill in the blanks. In evaluating the determinant below, about what row or column was it expanded? 5 1 1 87 4 97 6 1 ` 8 7 9 7 ` 4 ` 5 1 9 7 `  6 ` 5 1 8 7 `

Fill in the blanks. What is the denominator determinant for the system e ? 3x  4y 7 2x 3y 5

Fill in the blanks. What is the denominator determinant for the system x  2y 8 3x  y z 2 8x  4y z 6

Fill in the blanks. For the system e 3x  2y 1 4x y 3 Dx 7 Dy 5 D and D 11. Find the solution of the system.

Fill in the blanks. For the system 2x  3y z 8 x y z 2 4x  3y  z 6 Dx 28 Dy 14 Dz 14 D 14 and D 14. Find the solution of the system.

Fill in the blanks. Fill in the blank. If the denominator determinant for a system of equations is 0, the equations of the system are dependent or the system is _____.

Complete the evaluation of each determinant. ` 5 2 2 6 ` 5( ) (2)(2) D Dx 4 ` 26

Complete the evaluation of each determinant. 213 342 153

Evaluate each determinant. See Example 1. ` ` 2 3 2 5

Evaluate each determinant. See Example 1. ` 3 2 2 4 ` ` 2

Evaluate each determinant. See Example 1. ` ` 9 7 4 2 `

Evaluate each determinant. See Example 1. ` 1 2 3 4 `

Evaluate each determinant. See Example 1. ` ` 5 20 10 6

Evaluate each determinant. See Example 1. 10 15 15 5

Evaluate each determinant. See Example 1. ` ` 6 2 15 4 `

Evaluate each determinant. See Example 1. ` 3 2 12 8 `

Evaluate each determinant. See Example 1. ` ` 9 1 10 5 `

Evaluate each determinant. See Example 1. ` 7 7 6 4 ` `

Evaluate each determinant. See Example 1. ` ` 8 8 9 9 `

Evaluate each determinant. See Example 1. ` 20 3 20 3 `

Evaluate each determinant. See Examples 2 and 3. 321 412 531

Evaluate each determinant. See Examples 2 and 3. 623 154 235

Evaluate each determinant. See Examples 2 and 3. 1 2 3 2 11 3 2 1

Evaluate each determinant. See Examples 2 and 3. 11 2 2 1 2 31 3

Evaluate each determinant. See Examples 2 and 3. 251 034 126

Evaluate each determinant. See Examples 2 and 3. 4 1 2 6 1 0 1 3 4

Evaluate each determinant. See Examples 2 and 3. 1 4 1 3 0 2 3 1 2

Evaluate each determinant. See Examples 2 and 3. 8 3 1 1 02 3 9 4

Evaluate each determinant. See Examples 2 and 3. 12 1 37 3 4 3 5

Evaluate each determinant. See Examples 2 and 3. 147 258 369

Evaluate each determinant. See Examples 2 and 3. 120 012 001

Evaluate each determinant. See Examples 2 and 3. 101 010 111

Use Cramers rule to solve each system of equations. See Example 4. e x  y 6 x y 2

Use Cramers rule to solve each system of equations. See Example 4. e x y 4 2x  y 5 e

Use Cramers rule to solve each system of equations. See Example 4. e x  2y 21 x 2y 11

Use Cramers rule to solve each system of equations. See Example 4. e 5x  2y 11 7x  6y 9 e

Use Cramers rule to solve each system of equations. See Example 4. e 3x 4y 9 x  2y 8

Use Cramers rule to solve each system of equations. See Example 4. e 2x  2y 1 3x  4y 0 e

Use Cramers rule to solve each system of equations. See Example 4. e 2x  3y 31 3x  2y 39

Use Cramers rule to solve each system of equations. See Example 4. e 5x  3y 72 3x  5y 56 e

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. See Example 5. e 3x  2y 11 6x  4y 11

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. See Example 5. e 5x 4y 20 10x 8y 30 e

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. See Example 5. 5 6 x 2 y 10x  12y 24

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. See Example 5. 16x 8y 32 x 2 y 2 5

Use Cramers rule to solve each system of equations. See Example 6. x  y  z 4 x  y z 0 x y  z 2

Use Cramers rule to solve each system of equations. See Example 6. x  y  z 4 x y  z 2 x y z 0

Use Cramers rule to solve each system of equations. See Example 6. 3x  2y z 8 2x y  7z 10 2x  2y 3z 10

Use Cramers rule to solve each system of equations. See Example 6. x  2y  2z 10 2x  y  2z 9 2x  2y  z 1

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 2x  y  z 5 x 2y  3z 10 x  y 4z 3

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. x  y  2z 7 x  2y  z 8 2x  y  z 9

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e y 2x  1 3 3x 2y 8

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e 2x  3y 1 x y 9

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 4x 3y 1 6x 8z 1 2y 4z 0 e 2x

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 4x  3z 4 2y 6z 1 8x  4y  3z 9

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 2x  y z 1 0 x  2y  2z 2 0 4x  5y  3z 3 0

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 2x y  4z  2 0 5x  8y  7z 8 x  3y  z  3 0

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e 3x 16 5y 3x  5y 33 0

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e 2x  5y 13 0 2x  13 5y e

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. x  y 1 1 2 y  z 5 2 x z 3

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 1 2 x  y  z  3 2 0 x  1 2 y  z 1 2 0 x  y  1 2 z  1 2 0 x

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e 2x  3y 0 4x 6y 4

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. e 4x 3y 1 8x  3y 4 e 2

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. 2x  3y  4z 6 2x 3y 4z 4 4x  6y  8z 12 e 4x

Use Cramers rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. x 3y  4z 2 0 2x  y  2z 3 0 4x 5y  10z 7 0 2x 

Use a calculator with matrix capabilities. Evaluate each determinant. See Using Your Calculator: Evaluating Determinants. 25 36 44 11 21 54 37 31 19

Use a calculator with matrix capabilities. Evaluate each determinant. See Using Your Calculator: Evaluating Determinants. 13 27 62 38 27 52 10 300 42

Use a calculator with matrix capabilities. Evaluate each determinant. See Using Your Calculator: Evaluating Determinants. 280 191 356 211 102 422 400 213 333 1

Use a calculator with matrix capabilities. Evaluate each determinant. See Using Your Calculator: Evaluating Determinants. 4.1 2.2 3.3 2.7 5.9 6.8 2.3 5.3 0.6

Signaling. Solve the following system using Cramers rule to find the measures (in degrees) of the angles shown in the illustration. (Hint: Each equation must be written in standard Ax  By C form. e 2x  y 180 y x  30 A x x y

Investing. Solve the following system using Cramers rule to find the amount invested in each stock. (Hint: Each equation must be written in standard Ax  By  Cz D form.) x  y  z 20,000 0.10x  0.05y  0.06z 1,320 x 1 3 (y  z) Stock Amount Invested Rate of return HiTech x 10% SaveTel y 5% OilCo z 6%

Explain the difference between a matrix and a determinant. Give an example of each.

When evaluating 4 1 2 6 1 0 1 3 4 , why is it helpful to expand by the minors of the numbers in the third column?

Explain how to find the minor of an element of a determinant.

Explain how to find when solving a system of three linear equations in x, y, and z by Cramers rule. Use the words coefficients and constants in your explanation.

Explain how the following checkerboard pattern is used when evaluating a 3  3 determinant.   

Briefly describe each of the five methods of this chapter that can be used to solve a system of two linear equations in two variables.

Are the lines described by y 2x 7 and x 2y 7 perpendicular?

Are the lines described by y 2x 7 2x and 2x y 10y parallel?

How are the graphs of (x) x2 and g(x) x2 x related?

Is the graph of a circle the graph of a function?

The graph of a line passes through (0, 3). Is this the x-intercept or the y-intercept of the line?

What is the name of the function (x) 0 x 0?

For the function y 2x2  6x  1, what is the independent variable and what is the dependent variable?

If (x) x 3 x, what is (1) 3 ?

Show that x y 1 231 351 0 represents the equation of the line passing through (2, 3) and (3, 5).

Show that 1 2 001 301 041 represents the area of the triangle with vertices at (0, 0), (3, 0), and (0, 4) .

Use a check to determine whether the ordered pair is a solution of the system of equations. 11, 1 2 2, e x  2y 0 x  4y 1

Use a check to determine whether the ordered pair is a solution of the system of equations. (13, 23), e 3a 2b  7 0 2a  b 4 e (13

See the illustration. a. Give three points that satisfy the equation 2x  y 5. b. Give three points that satisfy the equation y x 4 2. c. Find the solution of: e 2x  y 5 y x 4 y 111 1 2 3 4 2 3 3 2 2345

Video Viewing. The graph below by the Diffusion Group (TDG) predicts American video viewing habits in the future. In what year does the graph predict that the weekly amount of time spent viewing live broadcast television will be the same as that spent viewing Internet video? Approximately how many hours of weekly viewing of each type will that be? 0 5 10 15 20 25 30 35 Year 2010 2012 2014 2016 2018 2020 2022

Solve each system by the graphing method, if possible. If a system is inconsistent or if the equations are dependent, state this. e 2x  y 11 x  2y 7

Solve each system by the graphing method, if possible. If a system is inconsistent or if the equations are dependent, state this. e y 3 2 x 2x 3y  13 0 e

Solve each system by the graphing method, if possible. If a system is inconsistent or if the equations are dependent, state this. e 1 2 x  1 3 y 2 y 6 3 2 x

Solve each system by the graphing method, if possible. If a system is inconsistent or if the equations are dependent, state this. e x 3 y 2 1 6x 9y 3

Solve each system using the substitution method. e x y 4 2x  3y 7

Solve each system using the substitution method. e y 2x  5 3x 5y 4 e

Solve each system using the elimination (addition) method. e x 2y 11 x  2y 21

Solve each system using the elimination (addition) method. e 4a  5b 9 6a 3b 3

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. 1 2 a 3 8 b 9 2 3 a 1 4 b 8 e 4a

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. y x 3 2 x 2y  7 2

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. e 4x 8y  5 8x 1 2y

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. e x  3y 2 2(x  3y) 4 e

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. e 0.07x 0.05  0.09y 7x 9y 8

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. e 0.1x  0.2y 1.1 2x y 2 e

Estimate the solution of the system e y 2 3 x 2x 3y 4 23. from the graphs in the illustration. Then solve the system algebraically.

Give one advantage and one disadvantage for each of the following methods for solving a system of two linear equations in two variables. a. Graphing method b. Substitution method c. Elimination method

Write a system of two equations in two variables to solve each problem. Library Cards. Residents of a city are charged $3 for a library card; nonresidents are charged $7. On a day when a total of 11 library cards were purchased, the receipts were $41. How many resident and nonresident library cards were sold?

Write a system of two equations in two variables to solve each problem. Maps. Refer to the illustration in the next column. The distance between Austin and Houston is 4 miles less than twice the distance between Austin and San Antonio. The round trip from Houston to Austin to San Antonio and back to Houston is 442 miles. Determine the mileages between Austin and Houston and between Austin and San Antonio. Mileage Map Houston Austin Victoria Corpus Christi Laredo San Del Antonio

Write a system of two equations in two variables to solve each problem. Riverboats. A Mississippi riverboat travels 30 miles downstream in three hours and then makes the return trip upstream in five hours. Find the speed of the riverboat in still water and the speed of the current.

Write a system of two equations in two variables to solve each problem. Mixing Solutions. How many fluid ounces of 6% sucrose solution must be mixed with 18% sucrose solution to make 750 ounces of a 10% sucrose solution?

Write a system of two equations in two variables to solve each problem. Investing. One year, a couple invested a total of $10,000 in two projects. The first investment, a mini-mall, made a 6% profit. The other investment, a skateboard park, made a 12% profit. If their investments made $960, how much was invested at each rate?

Write a system of two equations in two variables to solve each problem. Cooking. Two teaspoons and five tablespoons is a total of 85 milliliters of liquid. Five teaspoons and two tablespoons is a total of 55 milliliters of liquid. Find the number of milliliters in one teaspoon and the number of milliliters in one tablespoon.

Use a check to determine whether is a solution of the system: x y  z 4 x  2y z 1 x  y 3z 1

A system of three linear equations in three variables is graphed on the right. Does the system have a solution? If so, how many solutions does it have?

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  2y  3z 7 x 3y 2z 8 2x y z 7

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  y  z 4 x 2y z 1 2x y 2z 1 x 2

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  y z 3 x  z 2 2x y 3 2z

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. b 4c 2 a b  2c 1 2a 2b 2 5c x 

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  2z 10 3x  2y 3z 8 y  4z 6

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  3y  z 14 x 5y 19 3y  z 13 x

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x  3y  z 5 x  2y z 6 3x  y  2z 4 x 

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 3x  3y  6z 6 x y 2z 2 2x  2y  4z 4 2x 

Teddy Bears. A toy company produces three sizes of teddy bears. Each day, the total cost to produce the bears is $850, the total time needed to stuff them is 480 minutes, and the total time needed to sew them is 1,260 minutes. Use the information in the table to determine how many of each type of teddy bear are produced daily. Size of Production Stuffing Sewing teddy bear cost time time Small $3 2 min 6 min Medium $5 3 min 8 min Large $10 5 min 12 min

Financial Planning. A financial planner invested $22,000 in three accounts, paying 5%, 6%, and 7% annual interest. She invested $2,000 more at 6% than at 5%. If the total interest earned in one year was $1,370, how much was invested at each rate?

Ballistics. The path of a thrown object is a parabola with an equation of y ax2  bx  c. The parabola passes through the points (0, 0) (8, 12), and (12, 15). Find a, b, and c.

Veterinary Medicine. The daily requirements of a balanced diet for an animal are shown in the nutritional pyramid. The number of grams per cup of nutrients in three food mixes are shown in the table. How many cups of each mix should be used to meet the daily requirements for protein, carbohydrates, and essential fatty acids in the animals diet? Vitamins Minerals Essential fatty acids: 5 grams Carbohydrates: 10 grams Quality protein: 24 grams Grams per cup Protein Carbohydrates Fatty acids Mix A 52 1 Mix B 63 2 Mix C 83 1

Represent each system of equations using an augmented matrix. e 5x  4y 3 x y 3

Represent each system of equations using an augmented matrix. x  2y  3z 6 x 3y z 4 6x  y 2z 1

Perform each of the following elementary row operations on the augmented matrix c 6 12 6 1 3 2 d a. R1 4 R2 b. 1 6 R1 c. 6R2  R1

Perform each of the following elementary row operations on the augmented matrix: 2 113 110 1 3 1 2 7 676 a. R1 4 R2 b. 3R2 c. 2R2  R1

Solve each system using matrices, if possible. If a system is inconsistent or if the equations are dependent, state this. e x y 4 3x  7y 18

Solve each system using matrices, if possible. If a system is inconsistent or if the equations are dependent, state this. x  2y 3z 5 x  y  z 0 3x  4y  2z 1

Solve each system using matrices, if possible. If a system is inconsistent or if the equations are dependent, state this. e 16x 8y 32 2x  y 4

Solve each system using matrices, if possible. If a system is inconsistent or if the equations are dependent, state this. x  2y z 4 x  3y  4z 1 2x  4y 2z 3

Evaluate each determinant. 2 3 4 3 `

Evaluate each determinant. ` 3 4 5 6 ` ` 2 3

Evaluate each determinant. 1 2 1 2 1 3 1 2 2

Evaluate each determinant. 3 2 2 1 2 2 2 1 1

Use Cramers rule to solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. e 3x 4y 10 2x  3y 1

Use Cramers rule to solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. e 6x 4y 6 3x  2y 5 e 3x

Use Cramers rule to solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. x  2y  z 0 2x  y  z 3 x  y  2z 5

Use Cramers rule to solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. 2x  3y  z 2 x  3y  2z 7 x y z 7