Chapter 7

Determine whether the following statement is true or false. Explain. A solution of a system of linear inequalities is an ordered pair that is a solution of any one of the inequalities in the system.

Describe the graph of a linear system that has the given number of solutions. Sketch an example. No solution

Explain what it means to solve a linear system using the graph-and-check method.

When you use linear combinations to solve a linear system, what is the purpose of using multiplication as a first step?

What four steps do you use to solve a system of linear equations by the substitution method?

Graph the system of linear inequalities. $$ \begin{aligned} &y \geq-2 x+2\\\ &y \leq-1 \end{aligned} $$

Describe the graph of a linear system that has the given number of solutions. Sketch an example. Infinitely many solutions

When solving a system of linear equations, how do you decide which variable to isolate in Step 1 of the substitution method?

Graph the system of linear inequalities. $$ \begin{aligned} &y>x\\\ &x<1 \end{aligned} $$

Describe the graph of a linear system that has the given number of solutions. Sketch an example. Exactly one solution

Use the linear system below. $$ \begin{array}{l} {-x+y=-2} \\ {2 x+y=10} \end{array} $$ Write each equation in slope-intercept form.

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &3 x+5 y=25\\\ &2 x-6 y=12 \end{aligned} $$

use the following system of equations. $$ \begin{aligned}&3 x+2 y=7\\\ &5 x-y=3 \end{aligned} $$ Which equation would you use to solve for y? Explain why.

Graph the system of linear inequalities. $$ \begin{array}{r} {x+1>y} \\ {y \geq 0} \end{array} $$

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? Explain. $$\begin{array}{c} {2 x+y=5} \\ {-6 x-3 y=-15} \end{array}$$

Choose a method to solve the linear system. Explain your choice. $$ \begin{array}{r} {2 x+y=0} \\ {x+y=5} \end{array} $$

Use the linear system below. $$ \begin{array}{l} {-x+y=-2} \\ {2 x+y=10} \end{array} $$ Graph both equations in the same coordinate plane.

Describe the steps you would use to solve the system of equations using linear combinations. Then solve the system. Justify each step. \(x+3 y=6\) \(x-3 y=12\)

use the following system of equations. $$ \begin{aligned} &3 x+2 y=7\\\ &5 x-y=3 \end{aligned} $$ Substitute the expression for y into the other equation and solve for x.

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? Explain. $$\begin{aligned} &-6 x+2 y=4\\\ &-9 x+3 y=12 \end{aligned}$$

Use the linear system below. $$ \begin{array}{l} {-x+y=-2} \\ {2 x+y=10} \end{array} $$ Estimate the coordinates of the point of intersection.

Describe the steps you would use to solve the system of equations using linear combinations. Then solve the system. Justify each step. \(3 x-4 y=7\) \(2 x-y=3\)

In Exercises 6–8, use the following information. You have 2.65 dollar in your pocket. You have a total of 16 coins, with only quarters and dimes. Let q equal the number of quarters and d equal the number of dimes. $$ Complete: ?+?=16 $$

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? Explain. $$\begin{aligned} &2 x+y=7\\\ &3 x-y=-2 \end{aligned}$$

Use the linear system below. $$ \begin{array}{l} {-x+y=-2} \\ {2 x+y=10} \end{array} $$ Check the coordinates algebraically by substituting them into each equation of the original linear system.

Describe the steps you would use to solve the system of equations using linear combinations. Then solve the system. Justify each step. \(\begin{aligned} 2 y &=2 x-2 \\ 2 x+3 y &=12 \end{aligned}\)

Use substitution to solve the linear system. Justify each step. $$ \begin{aligned} &3 x+y=3\\\ &7 x+2 y=1 \end{aligned} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{aligned} &-x+y=7\\\ &2 x-2 y=-18 \end{aligned}$$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{l} {3 x-2 y=11} \\ {-x+6 y=7} \end{array} $$ $$ (5,2) $$

Use linear combinations to solve the linear system. Then check your solution. \(x+y=4\) \(x-y=-10\)

Use substitution to solve the linear system. Justify each step. $$ \begin{aligned} &2 x+y=4\\\ &-x+y=1 \end{aligned} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{array}{r} {-4 x+y=-8} \\ {-12 x+3 y=-24} \end{array}$$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {6 x-3 y=-15} \\ {2 x+y=-3} \end{array} \quad(-2,1) $$

Use linear combinations to solve the linear system. Then check your solution. \(a-b=8\) \(a+b=20\)

Use substitution to solve the linear system. Justify each step. $$ \begin{aligned} &3 x-y=0\\\ &5 y=15 \end{aligned} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{array}{l} {-4 x+y=-8} \\ {2 x-2 y=-14} \end{array}$$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {x+3 y=15} \\ {4 x+y=6} \end{array} \quad(3,-6) $$

Use linear combinations to solve the linear system. Then check your solution. \(2 x+y=4\) \(x-y=2\)

Tell which equation you would use to isolate a variable. Explain. $$ \begin{aligned} &2 x+y=-10\\\ &3 x-y=0 \end{aligned} $$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {-5x+y=19} \\ { x-7y=3} \end{array} \quad(-4,-1) $$

Use linear combinations to solve the linear system. Then check your solution. \(m+3 n=2\) \(-m+2 n=3\)

Tell which equation you would use to isolate a variable. Explain. $$ \begin{aligned} &m+4 n=30\\\ &m-2 n=0 \end{aligned} $$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {-15x+7y=1} \\ { 3x-y=1} \end{array} \quad(3,5) $$

Use linear combinations to solve the linear system. Then check your solution. \(p+4 q=23\) \(-p+q=2\)

Tell which equation you would use to isolate a variable. Explain. $$ \begin{array}{c} {5 c+3 d=11} \\ {5 c-d=5} \end{array} $$

Graph the system of linear inequalities. $$ \begin{aligned} &y \geq 0\\\ &x \geq-2 \end{aligned} $$

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {-2x+y=11} \\ { -x-9y=-15} \end{array} \quad(6,1) $$

Choose a solution method to solve the linear system. Explain your choice, but do not solve the system. $$ \begin{aligned} &6 x+y=2\\\ &9 x-y=5 \end{aligned} $$

Use linear combinations to solve the linear system. Then check your solution. \(3 v-2 w=1\) \(2 v+2 w=4\)

Tell which equation you would use to isolate a variable. Explain. $$ \begin{aligned} 3 x-2 y &=19 \\ x+y &=8 \end{aligned} $$