Solved: Pulley System A system of pulleys is loaded with128-pound and 32-pound weights

A system of equations in ________ form has a stair-step pattern with leading coefficients of 1.

A solution of a system of three linear equations in three unknowns can be written as an ________ ________, which has the form x, y, z.

The process used to write a system of linear equations in row-echelon form is called ________ elimination.

Interchanging two equations of a system of linear equations is a ________ ________ that produces an equivalent system.

A system of equations is called ________ when the number of equations differs from the number of variables in the system.

The equation is called the ________ equation, and it models the height of an object at time that is moving in a vertical line with a constant acceleration

In Exercises 710, determine whether each ordered triple is a solution of the system of equations. 6x 4x  y 2y    z 3z 5z    1 19 25

In Exercises 710, determine whether each ordered triple is a solution of the system of equations. 3x  4y  z  17 5x  y  2z  2 2x  3y  7z  21

In Exercises 710, determine whether each ordered triple is a solution of the system of equations.  4x  y 8x  6y 3x  y  z   z   0 74

In Exercises 710, determine whether each ordered triple is a solution of the system of equations. 4x  y  8z  6 y  z  0 4x  7y  6

In Exercises 1116, use back-substitution to solve the system of linear equations. 2x  y  5z  24 y  2z  6 z  8

In Exercises 1116, use back-substitution to solve the system of linear equations. 4x  3y  2z  21 6y  5z  8 z  2

In Exercises 1116, use back-substitution to solve the system of linear equations. 2x  y  3z  10 y  z  12 z  2

In Exercises 1116, use back-substitution to solve the system of linear equations. x  y  2z  22 3y  8z 9 z  3

In Exercises 1116, use back-substitution to solve the system of linear equations. 4x  2y y   z z z    8 4 11

In Exercises 1116, use back-substitution to solve the system of linear equations. 5x 3y  8z  5z z    22 10 4

In Exercises 17 and 18, perform the row operation and write the equivalent system. Add Equation 1 to Equation 2. What did this operation accomplish?

In Exercises 17 and 18, perform the row operation and write the equivalent system. Add times Equation 1 to Equation 3. What did this operation accomplish?

In Exercises 1924, solve the system of linear equations and check any solutions algebraically. 2x x   3y y   10 0

In Exercises 1924, solve the system of linear equations and check any solutions algebraically. 2x x   y y   0 7

In Exercises 1924, solve the system of linear equations and check any solutions algebraically. 3x x   y 2y   9 2

In Exercises 1924, solve the system of linear equations and check any solutions algebraically. x 5x   2y 4y   1 23

In Exercises 1924, solve the system of linear equations and check any solutions algebraically.  1.5x 0.3x   0.8y 0.2y   0.1 0.7

In Exercises 1924, solve the system of linear equations and check any solutions algebraically. 15 x x   12 y y   13 35

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  x 2x 3x   y y    z z z    7 9 10

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. x x   y 2y 3y    z 4z 4z    5 13 13

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x 5x  3y 3y   2z 4z  2  4  4

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x x x    4y 2y y    z 3z z    1 2 1

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x  y  z  x  2y  2z  3x  y  z  7 9 5

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 5x  3y  2z  3 2x  4y  z  7 x  11y  4z  3

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 3x  5y  5z  1 5x  2y  3z  0 7x  y  3z  0

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x  y  3z  1 2x  6y  8z  3 6x  8y  18z  5

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x  3y 4x  3y 8x  3y  z  3z  0  0  0

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 4x  3y  17z  0 5x  4y  22z  0 4x  2y  19z  0

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  x x 2x   y y    4z  10z  2z  1 10 5

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  2x  2y  6z  3x  2y  6z  x  y  5z  4 1 3

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 3x  3y  6z  6 x  2y  z  5 5x  8y  13z  7

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. x  2z  5 3x  y  z  1 6x  y  5z  16

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  x 4x  2y   5z z  2  0

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  x 3y 2z18 5x13y12z80

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. x  2y  7z  2x  y  z  3x  9y  36z  4 13 33

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x 4x 2x   y 3y    3z  2z  13z  4 10 8

In Exercises 2546, solve the system of linear equations and check any solutions algebraically.  2x 4x   3y 9y  z   2 7

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. 2x  3y  3z  7 4x  18y  15z  44

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. x 2x  2y 3y y   z 4z    3w w 2w  4  0  1  5

In Exercises 2546, solve the system of linear equations and check any solutions algebraically. x  y 2x  3y 3x  4y x  2y  z  z  z  w  w  2w  w  6  0  4  0

In Exercises 47 and 48, an object moving vertically is at the given heights at the specified times. Find the position equation for the object. At second, feet At seconds, feet At seconds, feet

In Exercises 47 and 48, an object moving vertically is at the given heights at the specified times. Find the position equation for the object. At second, feet At seconds, feet At seconds, feet

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 0, 0, 2, 2, 4, 0

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 0, 3, 1, 4, 2, 3

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 2, 0, 3, 1, 4, 0

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 1, 3, 2, 2, 3, 3

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 12 , 1, 1, 3, 2, 13

In Exercises 4954, find the equation of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 2, 3, 1, 0, 12, 3

In Exercises 5558, find the equation of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 0, 0, 5, 5, 10, 0

In Exercises 5558, find the equation of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 0, 0, 0, 6, 3, 3

In Exercises 5558, find the equation of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 3, 1, 2, 4, 6, 8

In Exercises 5558, find the equation of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 0, 0, 0, 2, 3, 0

Sports In Super Bowl I, on January 15, 1967, the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth 6, 1, and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. There were six times as many touchdowns as field goals. Find the numbers of touchdowns, extra-point kicks, and field goals scored. (Source: National Football League)

Agriculture A mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer B and fertilizer C. Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer A, five parts of fertilizer B, and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture?

Finance A small corporation borrowed $775,000 to expand its clothing line. Some of the money was borrowed at 8%, some at 9%, and some at 10%. How much was borrowed at each rate when the annual interest owed was $67,500 and the amount borrowed at 8% was four times the amount borrowed at 10%?

Advertising A health insurance company advertises on television, on radio, and in the local newspaper. The marketing department has an advertising budget of $42,000 per month. A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 ads per month and have as many television ads as radio and newspaper ads combined. How many of each type of ad can the department run each month?

In Exercises 63 and 64, find the values of and in the figure.

In Exercises 63 and 64, find the values of and in the figure.

Geometry The perimeter of a triangle is 180 feet. The longest side of the triangle is 9 feet shorter than twice the shortest side. The sum of the lengths of the two shorter sides is 30 feet more than the length of the longest side. Find the lengths of the sides of the triangle.

Acid Mixture A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20%, and 50%. How many liters of each solution will satisfy each condition? (a) Use 2 liters of the 50% solution. (b) Use as little as possible of the 50% solution. (c) Use as much as possible of the 50% solution

Electrical Network Applying Kirchhoffs Laws to the electrical network in the figure, the currents and are the solution of the system Find the currents.

Pulley System A system of pulleys is loaded with 128-pound and 32-pound weights (see figure). The tensions and in the ropes and the acceleration of the 32-pound weight are found by solving the system of equations where and are in pounds and is in feet per second squared. Solve this system.

To find the least squares regression parabola for a set of points you can solve the following system of linear equations for and In Exercises 69 and 70, the sums have been evaluated. Solve the given system for and to find the least squares regression parabola for the points. Use a graphing utility to confirm the result.  12 4c 9c 29c    9b 29b 99b    29a 99a 353a    20 70 254

To find the least squares regression parabola for a set of points you can solve the following system of linear equations for and In Exercises 69 and 70, the sums have been evaluated. Solve the given system for and to find the least squares regression parabola for the points. Use a graphing utility to confirm the result.  4c 40c 40b   40a 544a    19 12 160

Data Analysis: Stopping Distance In testing a new automobile braking system, engineers recorded the speed (in miles per hour) and the stopping distance (in feet) in the following table. (a) Find the least squares regression parabola for the data by solving the system. (b) Graph the parabola and the data on the same set of axes. (c) Use the model to estimate the stopping distance when the speed is 70 miles per hour.

A wildlife management team studied the reproductive rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females and the percent of females that had offspring the following year were recorded. The table shows the results. (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the percent of females that had offspring when there were 170 females. (d) Use the model to estimate the number of females when 40% of the females had offspring.

In Exercises 73 and 74, find values of and that satisfy the system. These systems arise in certain optimization problems in calculus, and is called a Lagrange multiplier. 2x  2x  0 2y    0 y  x2  0

In Exercises 73 and 74, find values of and that satisfy the system. These systems arise in certain optimization problems in calculus, and is called a Lagrange multiplier.  2  2y  2  0 2x  1    0 2x  y  100  0

True or False? In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer.The system is in row-echelon form.

True or False? In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations.

Think About It Are the following two systems of equations equivalent? Give reasons for your answer. Finding Systems of Linear Equations In Exercises 7982, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers.) , 4, 7 6, 12 , 74  5, 2, 1 3, 4, 2 x  3y  z  7y  4z  7y  4z  6 1 16  x  3y  z  6 2x  y  2z  1 3x  2y  z  2

HOW DO YOU SEE IT? The number of sides and the combined number of sides and diagonals for each of three regular polygons are shown below. Write a system of linear equations to find an equation of the form that represents the relationship between and for the three polygons.

In Exercises 7982, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers 3, 4, 2

In Exercises 7982, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers 5, 2, 1

In Exercises 7982, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers 6, 12 , 74 

In Exercises 7982, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers 32 , 4, 7