Chapter 7

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

Determine whether the following statement is true or false. 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. Explain.

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

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Which equation would you choose to solve for y? Why?

Explain what it means to solve a system of linear equations.

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

How is graphing a system of linear inequalities the same as graphing a system of linear equations? How is it different?

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Solve for \(y\) in the equation that you chose.

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

Graph the linear system below. Then decide if the ordered pair is a solution of the system. $$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$ $$ (-4,-2) $$

After you graph a system of linear inequalities, how can you use algebra to check whether the correct region is shaded?

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &x+y=300\\\ &x+3 y=18 \end{aligned} $$

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Substitute the expression into the other equation and solve for \(x .\)

Graph the linear system below. Then decide if the ordered pair is a solution of the system. $$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$ $$ (4,-2) $$

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

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &x+3 y=6\\\ &x-3 y=12 \end{aligned}$$

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Substitute the value of \(x\) into your equation from Exercise 2 . What is the solution of the linear system?

Explain how you can tell if the system of linear equations has a solution. Then solve the system. $$ \begin{aligned}&x-y=2\\\&4 x-4 y=8\end{aligned} $$

Graph the linear system below. Then decide if the ordered pair is a solution of the system. $$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$ $$ (-4,2) $$

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

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &x-3 y=0\\\ &x+10 y=13 \end{aligned}$$

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Explain how you can check the solution algebraically and graphically.

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

Graph the linear system below. Then decide if the ordered pair is a solution of the system. $$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$ $$ (4,2) $$

Graph the system of linear inequalities. $$y>-2 x+2\( \)y \leq-1$$

Use the following problem. The total cost of 10 gallons of regular gasoline and 15 gallons of premium gasoline is \(\$ 32.75 .\) Premium costs \(\$ .20\) more per gallon than regular. What is the cost per gallon of each type of gasoline? Write a verbal model for the problem.

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &3 x-4 y=7\\\ &2 x-y=3 \end{aligned}$$

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

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

Graph the system of linear inequalities. \(y>x\) \(x \leq 1\)

Use the following problem. The total cost of 10 gallons of regular gasoline and 15 gallons of premium gasoline is \(\$ 32.75 .\) Premium costs \(\$ .20\) more per gallon than regular. What is the cost per gallon of each type of gasoline? Assign labels to the verbal model.

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &2 y=-2+2 x\\\ &2 x+3 y=12 \end{aligned}$$

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

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

Use the graph-and-check method to solve the system of linear equations. $$ \begin{aligned} &y=2 x-1\\\ &y=x+1 \end{aligned} $$

Graph the system of linear inequalities. \(x+1>y\) \(y \geq 0\)

Use the following problem. The total cost of 10 gallons of regular gasoline and 15 gallons of premium gasoline is \(\$ 32.75 .\) Premium costs \(\$ .20\) more per gallon than regular. What is the cost per gallon of each type of gasoline? Write a linear system as an algebraic model.

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &2 x+y=4\\\ &x-y=2 \end{aligned}$$

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

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

Use the graph-and-check method to solve the system of linear equations. $$ \begin{aligned} &y=-2 x+3\\\ &y=x-3 \end{aligned} $$

Use the following problem. The total cost of 10 gallons of regular gasoline and 15 gallons of premium gasoline is \(\$ 32.75 .\) Premium costs \(\$ .20\) more per gallon than regular. What is the cost per gallon of each type of gasoline? Solve the system and answer the question.

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &a-b=8\\\ &a+b=20 \end{aligned}$$

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

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

Use the graph-and-check method to solve the system of linear equations. $$ \begin{aligned} &y=\frac{1}{2} x+2\\\ &y=-x+5 \end{aligned} $$

Solve the linear system using all three methods. $$ \begin{aligned} &x+y=2\\\ &6 x+y=2 \end{aligned} $$

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &y-2 x=0\\\ &6 y+2 x=0 \end{aligned}$$

Use substitution to solve the linear system. $$\begin{aligned} &x-y=0\\\ &x+y=2 \end{aligned}$$

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