True or False In Exercises 69 and 70, determine

In Exercises 16, determine whether the equation is linear in the variables x and y. 2x 3y = 4

In Exercises 16, determine whether the equation is linear in the variables x and y. 3x 4xy = 0

In Exercises 16, determine whether the equation is linear in the variables x and y. 3 y + 2 x 1 = 0

In Exercises 16, determine whether the equation is linear in the variables x and y. x2 + y2 = 4

In Exercises 16, determine whether the equation is linear in the variables x and y. 2 sin x y = 14

In Exercises 16, determine whether the equation is linear in the variables x and y. (cos 3)x + y = 16

In Exercises 710, find a parametric representation of the solution set of the linear equation. x 4y = 0

In Exercises 710, find a parametric representation of the solution set of the linear equation. 3x 1 2y = 9

In Exercises 710, find a parametric representation of the solution set of the linear equation. x + y + z = 1

In Exercises 710, find a parametric representation of the solution set of the linear equation. 12x1 + 24x2 36x3 = 12

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 2x + y = 4 x y = 2

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x + 3y = 2 x + 2y = 3

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x + 3x y = 1 3y = 4

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 1 2x 1 3y = 2x + 4 3y = 1

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 3x 2x + 5y = 7 y = 9

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x + 3y = 4x + 3y = 17 7

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 2x y = 5x y = 5 11

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x 5y = 21 6x + 5y = 21

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x + 3 4 + y 1 3 = 2x y = 1 12

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x 1 2 + y + 2 3 = 4 x 2y = 5

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 0.05x 0.03y = 0.07 0.07x + 0.02y = 0.16

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 0.2x 0.5y = 0.3x 0.4y = 27.8 68.7

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. x 4 + y 6 = 1 x y = 3

In Exercises 1124, graph the system of linear equations. Solve the system and interpret your answer. 2x 3 + 4x + y 6 = y = 2 3 4

In Exercises 2530, use backsubstitution to solve the system. x1 x2 = 2 x2 = 3

In Exercises 2530, use backsubstitution to solve the system. 2x1 4x2 = 6 3x2 = 9

In Exercises 2530, use backsubstitution to solve the system. x + y 2y + z = z = 1 2z = 0 3 0

In Exercises 2530, use backsubstitution to solve the system. x y 3y + = z = 4z = 5 11 8

In Exercises 2530, use backsubstitution to solve the system. 5x1 + 2x1 + 2x2 x2 + x3 = 0 = 0

In Exercises 2530, use backsubstitution to solve the system. x1 + x2 x2 + x3 = 0 = 0

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 3x 6x + y = 3 2y = 1

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 4x 8x + 5y = 10y = 3 14

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 2x 1 2x + 8y = 3 y = 0

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 9x 1 2x + 4y = 5 1 3y = 0

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 4x 0.8x 8y = 1.6y = 9 1.8

s In Exercises 3136, complete parts (a)(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? 14.7x + 44.1x 2.1y = 6.3y = 1.05 3.15

In Exercises 3756, solve the system of linear equations. x1 3x1 x2 = 2x2 = 0 1

In Exercises 3756, solve the system of linear equations. 3x + 2y = 6x + 4y = 2 14

In Exercises 3756, solve the system of linear equations. 3u + u + v = 240 3v = 240

In Exercises 3756, solve the system of linear equations. x1 2x2 = 0 6x1 + 2x2 = 0

In Exercises 3756, solve the system of linear equations. 9x 3y = 1 1 5x + 2 5y = 1 3

In Exercises 3756, solve the system of linear equations. 2 3x1 + 4x1 + 1 6x2 = 0 x2 = 0

In Exercises 3756, solve the system of linear equations. x 2 4 + y 1 3 = x 3y = 2 20

In Exercises 3756, solve the system of linear equations. x1 + 4 3 + x2 + 1 2 = 3x1 x2 = 1 2

In Exercises 3756, solve the system of linear equations. . 0.02x1 0.05x2 = 0.03x1 + 0.04x2 = 0.19 0.52

In Exercises 3756, solve the system of linear equations. 0.05x1 0.03x2 = 0.21 0.07x1 + 0.02x2 = 0.17

In Exercises 3756, solve the system of linear equations. x x 2x + y 2y z = 0 z = 6 z = 5

In Exercises 3756, solve the system of linear equations. x + x + 4x + y 3y y + + z = 2 2z = 8 = 4

In Exercises 3756, solve the system of linear equations. 3x1 x1 + 2x1 2x2 + x2 3x2 + 4x3 = 1 2x3 = 3 6x3 = 8

In Exercises 3756, solve the system of linear equations. 5x1 2x1 + x1 3x2 + 4x2 11x2 + 2x3 = 3 x3 = 7 4x3 = 3

In Exercises 3756, solve the system of linear equations. 2x1 4x1 2x1 + + x2 + 3x2 3x3 = 2x3 = 13x3 = 4 10 8

In Exercises 3756, solve the system of linear equations. . x1 4x1 2x1 2x2 2x2 + + 4x3 = x3 = 7x3 = 13 7 19

In Exercises 3756, solve the system of linear equations. x 5x 3y + 15y + 2z = 18 10z = 18

In Exercises 3756, solve the system of linear equations. x1 2x2 + 3x1 + 2x2 5x3 = x3 = 2 2

In Exercises 3756, solve the system of linear equations. x + 2x + 3x + x + y 3y 4y 2y + z + z z + + + w = 6 w = 0 2w = 4 w = 0

In Exercises 3756, solve the system of linear equations. x1 3x1 4x2 x2 2x2 + x3 3x3 + 2x4 = 1 x4 = 2 x4 = 0 = 4

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. 123.5x + 61.3y 32.4z = 54.7x 45.6y + 98.2z = 42.4x 89.3y + 12.9z = 262.74 197.4 33.66

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. 120.2x + 62.4y 36.5z = 56.8x 42.8y + 27.3z = 88.1x + 72.5y 28.5z = 258.64 71.44 225.88

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. x1 + 0.5x1 + 0.33x1 + 0.25x1 + 0.5x2 + 0.33x2 + 0.25x2 + 0.2x2 + 0.33x3 + 0.25x3 + 0.2x3 + 0.17x3 + 0.25x4 = 1.1 0.21x4 = 1.2 0.17x4 = 1.3 0.14x4 = 1.4

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. 0.1x 2.5y + 1.2z 2.4x + 1.5y 1.8z + 0.4x 3.2y + 1.6z 1.6x + 1.2y 3.2z + 0.75w = 0.25w = 1.4w = 0.6w = 108 81 148 143 .8 .2

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. 2x1 3 7x2 + 2 9x3 = 2 3x1 + 4 9x2 2 5x3 = 4 5x1 1 8x2 + 4 3x3 = 349 630 19 45 139 150

In Exercises 5762, use a software program or a graphing utility to solve the system of linear equations. 8x 1 7y + 1 6z 1 5w = 1 1 7x + 1 6y 1 5z + 1 4w = 1 1 6x 1 5y + 1 4z 1 3w = 1 1 5x + 1 4y 1 3z + 1 2w = 1

In Exercises 6366, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 4x + 3y + 17z = 0 5x + 4y + 22z = 0 4x + 2y + 19z = 0

In Exercises 6366, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 2x + 3y 4x + 3y 8x + 3y + = 0 z = 0 3z = 0

In Exercises 6366, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 5x + 10x + 5x + 5y 5y + 15y z = 0 2z = 0 9z = 0

In Exercises 6366, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 16x + 3y + z = 0 16x + 2y z = 0

Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 227 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 578 milligrams of vitamin C. How much vitamin C is in an eight-ounce glass of each type of juice?

Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts 1 2 hour after the first plane, but its speed is 80 kilometers per hour faster. Two hours after the first plane departs, the planes are 3200 kilometers apart. Find the airspeed of each plane.

True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A system of one linear equation in two variables is always consistent. (b) A system of two linear equations in three variables is always consistent. (c) If a linear system is consistent, then it has infinitely many solutions.

True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A linear system can have exactly two solutions. (b) Two systems of linear equations are equivalent when they have the same solution set. (c) A system of three linear equations in two variables is always inconsistent.

Find a system of two equations in two variables, x1 and x2, that has the solution set given by the parametric representation x1 = t and x2 = 3t 4, where t is any real number. Then show that the solutions to the system can also be written as x1 = 4 3 + t 3 and x2 = t

Find a system of two equations in three variables, x1, x2, and x3, that has the solution set given by the parametric representation x1 = t, x2 = s, and x3 = 3 + s t where s and t are any real numbers. Then show that the solutions to the system can also be written as x1 = 3 + s t, x2 = s, and x3 = t.

Substitution In Exercises 7376, solve the system of equations by first letting A = 1%x, B = 1%y, and C = 1%z. 12 x 12 y = 7 3 x + 4 y = 0

Substitution In Exercises 7376, solve the system of equations by first letting A = 1%x, B = 1%y, and C = 1%z. 3 x + 2 y = 2 x 3 y = 1 17 6

Substitution In Exercises 7376, solve the system of equations by first letting A = 1%x, B = 1%y, and C = 1%z. 2 x 4 x 2 x + 1 y + 3 y 3 z = + 2 z = 13 z = 4 10 8

Substitution In Exercises 7376, solve the system of equations by first letting A = 1%x, B = 1%y, and C = 1%z. 2 x + 1 y 3 x 4 y 2 x + 1 y 2 z = = + 3 z = 5 1 0

Trigonometric Coefficients In Exercises 77 and 78, solve the system of linear equations for x and y. (cos )x + (sin )y = 1 (sin )x + (cos )y = 0

Trigonometric Coefficients In Exercises 77 and 78, solve the system of linear equations for x and y. (cos )x + (sin )y = 1 (sin )x + (cos )y = 1

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. No solution x + kx + ky = 2 y = 4

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Exactly one solution x + kx + ky = 0 y = 0

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. . Exactly one solutionkx + x + 2x 2ky + y + y + 3kz = z = z = 4k 0 1

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. No solution x + 2y + kz = 6 3x + 6y + 8z = 4

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Infinitely many solutions 4x + kx + ky = y = 6 3

Coefficient Design In Exercises 7984, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Infinitely many solutions kx + 3x y = 4y = 16 64

Determine the values of k such that the system of linear equations does not have a unique solution. x + x + kx + y + ky + y + kz = 3 z = 2 z = 1

CAPSTONE Find values of a, b, and c such that the system of linear equations has (a) exactly one solution, (b) infinitely many solutions, and (c) no solution. Explain. x + 5y + x + 6y 2x + ay + z = 0 z = 0 bz = c

Writing Consider the system of linear equations in x and y. a1x + b1y = c1 a2x + b2y = c2 a3x + b3y = c3 Describe the graphs of these three equations in the xy-plane when the system has (a) exactly one solution, (b) infinitely many solutions, and (c) no solution.

Writing Explain why the system of linear equations in Exercise 87 must be consistent when the constant terms c1, c2, and c3 are all zero.

Show that if ax2 + bx + c = 0 for all x, then a = b = c = 0.

Consider the system of linear equations in x and y. ax + cx + by = dy = e f Under what conditions will the system have exactly one solution?

Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? x 4y = 5x 6y = 3 13

Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? 2x 4x + 3y = 6y = 7 14

Writing In Exercises 93 and 94, the graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. 100y x = 99y x = 200 198

Writing In Exercises 93 and 94, the graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. . 21x 20y = 13x 12y = 0 120

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \(2 x-3 y=4\) Text Transcription: 2 x-3 y=4

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \(3 x-4 x y=0\) Text Transcription: 3 x-4 x y=0

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \(\frac{3}{y}+\frac{2}{x}-1=0\) Text Transcription: 3/y+2/x-1=0

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \(x^{2}+y^{2}=4\) Text Transcription: x^2+y^2=4

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \(2 \sin x-y=14\) Text Transcription: 2 sin x-y=14

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. \((\cos 3) x+y=-16\) Text Transcription: (cos 3) x+y=-16

Parametric Representation In Exercises 7-10, find a parametric representation of the solution set of the linear equation. \(2 x-4 y=0\) Text Transcription: 2 x-4 y=0

Parametric Representation In Exercises 7-10, find a parametric representation of the solution set of the linear equation. \(3 x-\frac{1}{2} y=9\) Text Transcription: 3 x-1/2 y=9

Parametric Representation In Exercises 7-10, find a parametric representation of the solution set of the linear equation. \(x+y+z=1\) Text Transcription: x+y+z=1

Parametric Representation In Exercises 7-10, find a parametric representation of the solution set of the linear equation. \(12 x_{1}+24 x_{2}-36 x_{3}=12\) Text Transcription: 12 x_1+24 x_2-36 x_3=12

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{r} 2 x+y=4 \\ x-y=2 \end{array} \) Text Transcription: 2 x+y=4 x-y=2

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{r} x+3 y=2 \\ -x+2 y=3 \end{array} \) Text Transcription: x+3 y=2 -x+2 y=3

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{r} -x+y=1 \\ 3 x-3 y=4 \end{array} \) Text Transcription: -x+y=1 3 x-3 y=4

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{rr} \frac{1}{2} x-\frac{1}{3} y & =1 \\ -2 x+\frac{4}{3} y & =-4 \end{array} \) Text Transcription: 1/2/-1/3y=1 -2x+4/3y=-4

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} 3 x-5 y=7 \\ 2 x+y=9 \end{array} \) Text Transcription: 3 x-5 y=7 2 x+y=9

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{r} -x+3 y=17 \\ 4 x+3 y=7 \end{array} \) Text Transcription: -x+3 y=17 4 x+3 y=7

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} 2 x-y=5 \\ 5 x-y=11 \end{array} \) Text Transcription: 2 x-y=5 5 x-y=11

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{r} x-5 y=21 \\ 6 x+5 y=21 \end{array} \) Text Transcription: x-5 y=21 6 x+5 y=21

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\frac{x+3}{4}+\frac{y-1}{3}=1\) Text Transcription: x+3/4+y-1/3=1

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\frac{x-1}{2}+\frac{y+2}{3}=4\) Text Transcription: x-1/2+y+2/3=4

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} 0.05 x-0.03 y=0.07 \\ 0.07 x+0.02 y=0.16 \end{array} \) Text Transcription: 0.05 x-0.03 y=0.07 0.07 x+0.02 y=0.16

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} 0.2 x-0.5 y=-27.8 \\ 0.3 x-0.4 y=68.7 \end{array} \) Text Transcription: 0.2 x-0.5 y=-27.8 0.3 x-0.4 y=68.7

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} \frac{x}{4}+\frac{y}{6}=1 \\ x-y=3 \end{array} \) Text Transcription: x/4+y/6=1 x-y=3

Graphical Analysis In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. \(\begin{array}{l} \frac{2 x}{3}+\frac{y}{6}=\frac{2}{3} \\ 4 x+y=4 \end{array} \) Text Transcription: 2x/3+y/6=2/3 4x+y=4

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{array}{r} x_{1}-x_{2}=2 \\ x_{2}=3 \end{array} \) Text Transcription: x_1-x_2=2 x_2=3

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{aligned} 2 x_{1}-4 x_{2} &=6 \\ 3 x_{2} &=9 \end{aligned} \) Text Transcription: 2 x_1-4 x_2 =6 3 x_2 =9

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{array}{r} -x+y-z=0 \\ 2 y+z=3 \\ \frac{1}{2} z=0 \end{array} \) Text Transcription: -x+y-z=0 2 y+z=3 1/2 z=0

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{aligned} x-y &=5 \\ 3 y+z &=11 \\ 4 z &=8 \end{aligned} \) Text Transcription: x-y =5 3 y+z =11 4 z =8

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{aligned} 5 x_{1}+2 x_{2}+x_{3} &=0 \\ 2 x_{1}+x_{2} &=0 \end{aligned} \) Text Transcription: 5 x_1+2 x_2+x_3 =0 2 x_1+x_2 =0

Back-Substitution In Exercises 25-30, use back substitution to solve the system \(\begin{aligned} x_{1}+x_{2}+x_{3} &=0 \\ x_{2} &=0 \end{aligned} \) Text Transcription: x_1+x_2+x_3 =0 x_2 =0

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{array}{r} -3 x-y=3 \\ 6 x+2 y=1 \end{array} \) Text Transcription: -3 x-y=3 6 x+2 y=1

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{array}{r} 4 x-5 y=3 \\ -8 x+10 y=14 \end{array} \) Text Transcription: 4 x-5 y=3 -8 x+10 y=14

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{array}{l} 2 x-8 y=3 \\ \frac{1}{2} x+y=0 \end{array} \) Text Transcription: 2 x-8 y=3 1/2 x+y=0

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{array}{l} 9 x-4 y=5 \\ \frac{1}{2} x+\frac{1}{3} y=0 \end{array} \) Text Transcription: 9 x-4 y=5 1/2 x+1/3 y=0

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{array}{r} 4 x-8 y=9 \\ 0.8 x-1.6 y=1.8 \end{array} \) Text Transcription: 4 x-8 y=9 0.8 x-1.6 y=1.8

Graphical Analysis In Exercises 31-36, complete parts (a)-(e) for the system of equations. (a) Use a graphing utility to graph the system. (b) Use the graph to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? \(\begin{aligned} -14.7 x+2.1 y &=1.05 \\ 44.1 x-6.3 y &=-3.15 \end{aligned} \) Text Transcription: 4 x-8 y=9 0.8 x-1.6 y=1.8

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} x_{1}-x_{2} &=0 \\ 3 x_{1}-2 x_{2} &=-1 \end{aligned} \) Text Transcription: x_1-x_2 =0 3 x_1-2 x_2 =-1

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{l} 3 x+2 y=2 \\ 6 x+4 y=14 \end{array} \) Text Transcription: 3 x+2 y=2 6 x+4 y=14

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} 3 u+v=240 \\ u+3 v=240 \end{array} \) Text Transcription: 3 u+v=240 u+3 v=240

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} x_{1}-2 x_{2}=0 \\ 6 x_{1}+2 x_{2}=0 \end{array} \) Text Transcription: x_1-2 x_2=0 6 x_1+2 x_2=0

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} 9 x-3 y=-1 \\ \frac{1}{5} x+\frac{2}{5} y=-\frac{1}{3} \end{array} \) Text Transcription: 9 x-3 y=-1 1/5 x+2/5 y=-13

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{l} \frac{2}{3} x_{1}+\frac{1}{6} x_{2}=0 \\ 4 x_{1}+x_{2}=0 \end{array} \) Text Transcription: 2/3 x_1+1/6 x_2=0 4 x_1+x_2=0

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} \frac{x-2}{4}+\frac{y-1}{3}=2 \\ x-3 y=20 \end{array} \) Text Transcription: x-2/4+y-1/3=2 x-3 y=20

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} \frac{x_{1}+4}{3}+\frac{x_{2}+1}{2} &=1 \\ 3 x_{1}-x_{2} &=-2 \end{aligned} \) Text Transcription: x_1+4/3+x_2+1/2 &=1 3 x_1-x_2 &=-2

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{l} 0.02 x_{1}-0.05 x_{2}=-0.19 \\ 0.03 x_{1}+0.04 x_{2}=0.52 \end{array} \) Text Transcription: 0.02 x_1-0.05 x_2=-0.19 0.03 x_1+0.04 x_2=0.52

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{l} 0.05 x_{1}-0.03 x_{2}=0.21 \\ 0.07 x_{1}+0.02 x_{2}=0.17 \end{array} \) Text Transcription: 0.05 x_1-0.03 x_2=0.21 0.07 x_1+0.02 x_2=0.17

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} x-y-z=0 \\ x+2 y-z=6 \\ 2 x-z=5 \end{array} \) Text Transcription: x-y-z=0 x+2 y-z=6 2 x-z=5

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} x+y+z &=2 \\ -x+3 y+2 z &=8 \\ 4 x+y &=4 \end{aligned} \) Text Transcription: x+y+z =2 -x+3 y+2 z =8 4 x+y =4

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} 3 x_{1}-2 x_{2}+4 x_{3}=1 \\ x_{1}+x_{2}-2 x_{3}=3 \\ 2 x_{1}-3 x_{2}+6 x_{3}=8 \end{array} \) Text Transcription: 3 x_1-2 x_2+4 x_3=1 x_1+x_2-2 x_3=3 2 x_1-3 x_2+6 x_3=8

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} 5 x_{1}-3 x_{2}+2 x_{3} &=3 \\ 2 x_{1}+4 x_{2}-x_{3} &=7 \\ x_{1}-11 x_{2}+4 x_{3} &=3 \end{aligned} \) Text Transcription: 5 x_1-3 x_2+2 x_3 =3 2 x_1+4 x_2-x_3 =7 x_1-11 x_2+4 x_3 =3

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} 2 x_{1}+x_{2}-3 x_{3} &=4 \\ 4 x_{1}+2 x_{3} &=10 \\ -2 x_{1}+3 x_{2}-13 x_{3} &=-8 \end{aligned} \) Text Transcription: 2 x_1+x_2-3 x_3 =4 4 x_1+2 x_3 =10 -2 x_1+3 x_2-13 x_3 =-8

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} x_{1}+4 x_{3} &=13 \\ 4 x_{1}-2 x_{2}+x_{3} &=7 \\ 2 x_{1}-2 x_{2}-7 x_{3} &=-19 \end{aligned} \) Text Transcription: x_1+4 x_3 =13 4 x_1-2 x_2+x_3 =7 2 x_1-2 x_2-7 x_3 =-19

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} x-3 y+2 z=18 \\ 5 x-15 y+10 z=18 \end{array} \) Text Transcription: x-3 y+2 z=18 5 x-15 y+10 z=18

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} x_{1}-2 x_{2}+5 x_{3} &=2 \\ 3 x_{1}+2 x_{2}-x_{3} &=-2 \end{aligned} \) Text Transcription: x_1-2 x_2+5 x_3 =2 3 x_1+2 x_2-x_3 =-2

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{array}{r} x+y+z+w=6 \\ 2 x+3 y-w=0 \\ -3 x+4 y+z+2 w=4 \\ x+2 y-z+w=0 \end{array} \) Text Transcription: x+y+z+w=6 2 x+3 y-w=0 -3 x+4 y+z+2 w=4 x+2 y-z+w=0

System of Linear Equations In Exercises 37-56, solve the system of linear equations. \(\begin{aligned} -x_{1}+2 x_{4} &=1 \\ 4 x_{2}-x_{3}-x_{4} &=2 \\ x_{2}-x_{4} &=0 \\ 3 x_{1}-2 x_{2}+3 x_{3} &=4 \end{aligned} \) Text Transcription: -x_1+2 x_4 =1 4 x_2-x_3-x_4 =2 x_2-x_4 =0 3 x_1-2 x_2+3 x_3 =4

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{aligned} 123.5 x+61.3 y-32.4 z &=&-262.74 \\ 54.7 x-45.6 y+98.2 z &=& & 197.4 \\ 42.4 x-89.3 y+12.9 z &=& 33.66 \end{aligned} \) Text Transcription: 123.5 x+61.3 y-32.4 z =-262.74 54.7 x-45.6 y+98.2 z = 197.4 42.4 x-89.3 y+12.9 z = 33.66

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{aligned} 120.2 x+62.4 y-36.5 z &=258.64 \\ 56.8 x-42.8 y+27.3 z &=-71.44 \\ 88.1 x+72.5 y-28.5 z &=225.88 \end{aligned} \) Text Transcription: 120.2 x+62.4 y-36.5 z =258.64 56.8 x-42.8 y+27.3 z =-71.44 88.1 x+72.5 y-28.5 z =225.88

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{aligned} x_{1}+0.5 x_{2}+0.33 x_{3}+0.25 x_{4} &=1.1 \\ 0.5 x_{1}+0.33 x_{2}+0.25 x_{3}+0.21 x_{4} &=1.2 \\ 0.33 x_{1}+0.25 x_{2}+0.2 x_{3}+0.17 x_{4} &=1.3 \\ 0.25 x_{1}+0.2 x_{2}+0.17 x_{3}+0.14 x_{4} &=1.4 \end{aligned} \) Text Transcription: x_1+0.5 x_2+0.33 x_3+0.25 x_4 =1.1 0.5 x_1+0.33 x_2+0.25 x_3+0.21 x_4 =1.2 0.33 x_1+0.25 x_2+0.2 x_3+0.17 x_4 =1.3 0.25 x_1+0.2 x_2+0.17 x_3+0.14 x_4 =1.4

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{array}{lrl} 0.1 x-2.5 y+1.2 z-0.75 w= & 108 \\ 2.4 x+1.5 y-1.8 z+0.25 w= & -81 \\ 0.4 x-3.2 y+1.6 z-1.4 w= & 148.8 \\ 1.6 x+1.2 y-3.2 z+0.6 w= & -143.2 \end{array} \) Text Transcription: 0.1 x-2.5 y+1.2 z-0.75 w= 108 2.4 x+1.5 y-1.8 z+0.25 w= -81 0.4 x-3.2 y+1.6 z-1.4 w= 148.8 1.6 x+1.2 y-3.2 z+0.6 w= -143.2

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{array}{l} \frac{1}{2} x_{1}-\frac{3}{7} x_{2}+\frac{2}{9} x_{3}=\frac{349}{630} \\ \frac{2}{3} x_{1}+\frac{4}{9} x_{2}-\frac{2}{5} x_{3}=-\frac{19}{45} \\ \frac{4}{5} x_{1}-\frac{1}{8} x_{2}+\frac{4}{3} x_{3}=\frac{139}{150} \end{array} \) Text Transcription: 1/2 x_1-3/7 x_2+2/9 x_3=349/630 2/3 x_1+4/9 x_2-2/5 x_3=-19/45 4/5 x_1-1/8 x_2+4/3 x_3=139/150

System of Linear Equations In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. \(\begin{array}{l} \frac{1}{8} x-\frac{1}{7} y+\frac{1}{6} z-\frac{1}{5} w=1 \\ \frac{1}{7} x+\frac{1}{6} y-\frac{1}{5} z+\frac{1}{4} w=1 \\ \frac{1}{6} x-\frac{1}{5} y+\frac{1}{4} z-\frac{1}{3} w=1 \\ \frac{1}{5} x+\frac{1}{4} y-\frac{1}{3} z+\frac{1}{2} w=1 \end{array} \) Text Transcription: 1/8 x-1/7 y+1/6 z-1/5 w=1 1/7 x+1/6 y-1/5 z+1/4 w=1 1/6 x-1/5 y+1/4 z-1/3 w=1 1/5 x+1/4 y-1/3 z+1/2 w=1

Number of Solutions In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. \(\begin{array}{l} 4 x+3 y+17 z=0 \\ 5 x+4 y+22 z=0 \\ 4 x+2 y+19 z=0 \end{array} \) Text Transcription: 4 x+3 y+17 z=0 5 x+4 y+22 z=0 4 x+2 y+19 z=0

Number of Solutions In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. \(\begin{array}{l} 2 x+3 y=0 \\ 4 x+3 y-z=0 \\ 8 x+3 y+3 z=0 \end{array} \) Text Transcription: 2 x+3 y=0 4 x+3 y-z=0 8 x+3 y+3 z=0

Number of Solutions In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. \(\begin{array}{r} 5 x+5 y-z=0 \\ 10 x+5 y+2 z=0 \\ 5 x+15 y-9 z=0 \end{array} \) Text Transcription: 5 x+5 y-z=0 10 x+5 y+2 z=0 5 x+15 y-9 z=0

Number of Solutions In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. \(\begin{array}{l} 16 x+3 y+z=0 \\ 16 x+2 y-z=0 \end{array} \) Text Transcription: 16 x+3 y+z=0 16 x+2 y-z=0

Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 227 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 578 milligrams of vitamin C. How much vitamin C is in an eight-ounce glass of each type of juice?

Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts 1 2 hour after the first plane, but its speed is 80 kilometers per hour faster. Two hours after the first plane departs, the planes are 3200 kilometers apart. Find the airspeed of each plane.

True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A system of one linear equation in two variables is always consistent. (b) A system of two linear equations in three variables is always consistent. (c) If a linear system is consistent, then it has infinitely many solutions.

True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A linear system can have exactly two solutions. (b) Two systems of linear equations are equivalent when they have the same solution set. (c) A system of three linear equations in two variables is always inconsistent.

Find a system of two equations in two variables, \(x_{1}\) and \(x_{2}\), that has the solution set given by the parametric representation \(x_{1}=t\) and \(x_{2}=3 t-4\), where t is any real number. Then show that the solutions to the system can also be written as \(x_{1}=\frac{4}{3}+\frac{t}{3}\) and \(x_{2}=t\) Text Transcription: x_1 x_2 x_1=t x_2=3t-4 x_1=4/3+t/3 x_2=t

Find a system of two equations in three variables, \(x_{1},x_{2}\) and \(x_{3}\), that has the solution set given by the parametric representation \(x_{1}=t, \quad x_{2}=s\), and \(x_{3}=3+s-t\) where s and t are any real numbers. Then show that the solutions to the system can also be written as \(x_{1}=3+s-t, \quad x_{2}=s\), and \(x_{3}=t\). Text Transcription: x_1,x_2 x_3 x_1=t, x_2=s x_3=3+s-t x_1=3+s-t x_2=s x_3=t

Substitution In Exercises 73–76, solve the system of equations by first letting A = 1/x, B = 1/y, and C = 1/z. \(\begin{array}{c} \frac{12}{x}-\frac{12}{y}=7 \\ \frac{3}{x}+\frac{4}{y}=0 \end{array} \) Text Transcription: 12/x-12/y=7 3/x+4/y=0

Substitution In Exercises 73–76, solve the system of equations by first letting A = 1/x, B = 1/y, and C = 1/z. \(\begin{array}{l} \frac{3}{x}+\frac{2}{y}=-1 \\ \frac{2}{x}-\frac{3}{y}=-\frac{17}{6} \end{array} \) Text Transcription: 3/x+2/y=-1 2/x-3/y=-17/6

Substitution In Exercises 73–76, solve the system of equations by first letting A = 1/x, B = 1/y, and C = 1/z. \(\begin{array}{c} \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=4 \\ \frac{4}{x}+\frac{2}{z}=10 \\ -\frac{2}{x}+\frac{3}{y}-\frac{13}{z}=-8 \end{array} \) Text Transcription: 2/x+1/y-3/z=4 4/x+2/z=10 -2/x+3/y-13/z=-8

Substitution In Exercises 73–76, solve the system of equations by first letting A = 1/x, B = 1/y, and C = 1/z. \(\begin{array}{l} \frac{2}{x}+\frac{1}{y}-\frac{2}{z}=5 \\ \frac{3}{x}-\frac{4}{y}=-1 \\ \frac{2}{x}+\frac{1}{y}+\frac{3}{z}=0 \end{array} \) Text Transcription: 2/x+1/y-2/z=5 3/x-4/y=-1 2/x+1/y+3/z=0

Trigonometric Coefficients In Exercises 77 and 78, solve the system of linear equations for x and y. \(\begin{array}{r} (\cos \theta) x+(\sin \theta) y=1 \\ (-\sin \theta) x+(\cos \theta) y=0 \end{array} \) Text Transcription: (cos theta) x+(sin theta) y=1 (-sin theta) x+(cos theta) y=0

Trigonometric Coefficients In Exercises 77 and 78, solve the system of linear equations for x and y. \(\begin{array}{r} (\cos \theta) x+(\sin \theta) y=1 \\ (-\sin \theta) x+(\cos \theta) y=1 \end{array} \) Text Transcription: (cos theta) x+(sin theta) y=1 (-sin theta) x+(cos theta) y=1

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. No solution x + ky = 2 kx + y = 4

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Exactly one solution x + ky = 0 kx + y = 0

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Exactly one solution kx + 2ky + 3kz = 4k x + y + z = 0 2x - y + z = 1

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. No solution x + 2y + kz = 6 3x + 6y + 8z = 4

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Infinitely many solutions 4x + ky =6 kx + y = -3

Coefficient Design In Exercises 79-84, determine the value(s) of k such that the system of linear equations has the indicated number of solutions. Infinitely many solutions kx + y = 16 3x - 4y = - 64

Determine the values of k such that the system of linear equations does not have a unique solution. x + y + kz = 3 x + ky + z = 2 kx + y + = 1

CAPSTONE Find values of a, b, and c such that the system of linear equations has (a) exactly one solution, (b) infinitely many solutions, and (c) no solution. Explain. x + 5y + z = 0 x + 6y - z = 0 2x + ay + bz = c

Writing Consider the system of linear equations in x and y. \(\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\a_{2} x+b_{2} y=c_{2} \\a_{3} x+b_{3} y=c_{3}\end{array}\) Describe the graphs of these three equations in the xy-plane when the system has (a) exactly one solution, (b) infinitely many solutions, and (c) no solution. Text Transcription: a_1 x+b_1 y=c_1 a_2 x+b_2 y=c_2 a_3 x+b_3 y=c_3

Writing Explain why the system of linear equations in Exercise 87 must be consistent when the constant terms \(c_{1}, c_{2}\), and \(c_{2}\) are all zero. Text Transcription: c_1, c_2 c_3

Show that if \(a x^{2}+b x+c=0\) for all x, then a = b =c = 0. Text Transcription: a x^2+b x+c=0

Consider the system of linear equations in x and y. ax + by = e cx + dy = f Under what conditions will the system have exactly one solution?

Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? x - 4y = -3 5x - 6y= 13

Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? 2x - 3y = 7 -4x + 6y = -14

Writing In Exercises 93 and 94, the graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. 100y - x = 200 99y - x = -198

Writing In Exercises 93 and 94, the graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. . 21x 20y = 13x 12y = 0 120