5.1 Solve Systems of Equations by Graphing

Before you get started, take this readiness quiz.

For the equation $y = \frac{2}{3} x - 4$
ⓐ is $\left(6 , 0\right)$ a solution? ⓑ is $\left(−3 , −2\right)$ a solution?
If you missed this problem, review the reference.

Find the slope and y-intercept of the line $3 x - y = 12$.
If you missed this problem, review the reference.

Find the x- and y-intercepts of the line $2 x - 3 y = 12$.
If you missed this problem, review the reference.

Determine whether the ordered pair is a solution to the system: $\begin{cases} 3 x + y = 0 \\ x + 2 y = −5 \end{cases} .$

ⓐ $\left(1 , −3\right)$ ⓑ $\left(0 , 0\right)$

Determine whether the ordered pair is a solution to the system: $\begin{cases} x - 3 y = −8 \\ −3 x - y = 4 \end{cases} .$

ⓐ $\left(2 , −2\right)$ ⓑ $\left(−2 , 2\right)$

Solve the system by graphing: $\begin{cases} x - 3 y = −3 \\ x + y = 5 \end{cases} .$

Solve the system by graphing: $\begin{cases} - x + y = 1 \\ 3 x + 2 y = 12 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = 2 x + 2 \\ y = − x - 4 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = 3 x + 3 \\ y = − x + 7 \end{cases} .$

Solve each system by graphing: $\begin{cases} - x + y = 1 \\ 2 x + y = 10 \end{cases} .$

Solve each system by graphing: $\begin{cases} 2 x + y = 6 \\ x + y = 1 \end{cases} .$

Solve each system by graphing: $\begin{cases} x + y = 6 \\ x - y = 2 \end{cases} .$

Solve each system by graphing: $\begin{cases} x + y = 2 \\ x - y = −8 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = −1 \\ x + 3 y = 6 \end{cases} .$

Solve each system by graphing: $\begin{cases} x = 4 \\ 3 x - 2 y = 24 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = - \frac{1}{4} x + 2 \\ x + 4 y = - 8 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = 3 x - 1 \\ 6 x - 2 y = 6 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = - 3 x - 6 \\ 6 x + 2 y = - 12 \end{cases} .$

Solve each system by graphing: $\begin{cases} y = \frac{1}{2} x - 4 \\ 2 x - 4 y = 16 \end{cases} .$

Without graphing, determine the number of solutions and then classify the system of equations: $\begin{cases} y = 3 x - 1 \\ 6 x - 2 y = 12 \end{cases} .$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = - 2 x - 4 \\ 4 x + 2 y = 9 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = \frac{1}{3} x - 5 \\ x - 3 y = 6 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations: $\begin{cases} 2 x + y = - 3 \\ x - 5 y = 5 \end{cases} .$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} 3 x + 2 y = 2 \\ 2 x + y = 1 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} x + 4 y = 12 \\ - x + y = 3 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations. $\begin{cases} 3 x - 2 y = 4 \\ y = \frac{3}{2} x - 2 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} 4 x - 5 y = 20 \\ y = \frac{4}{5} x - 4 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations.

$\begin{cases} −2 x - 4 y = 8 \\ y = - \frac{1}{2} x - 2 \end{cases}$

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.

$\begin{cases} 2 x - 6 y = 0 \\ 3 x - 4 y = 5 \end{cases}$

ⓐ $\left(3 , 1\right)$ ⓑ $\left(−3 , 4\right)$

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.

$\begin{cases} 2 x + y = 5 \\ x + y = 1 \end{cases}$

ⓐ $\left(4 , −\text{3}\right)$ ⓑ $\left(2 , 0\right)$

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.

$\begin{cases} x + y = 2 \\ y = \frac{3}{4} x \end{cases}$

ⓐ $\left(\frac{8}{7} , \frac{6}{7}\right)$ ⓑ $\left(1 , \frac{3}{4}\right)$

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.

$\begin{cases} x + 5 y = 10 \\ y = \frac{3}{5} x + 1 \end{cases}$

ⓐ $\left(−10 , 4\right)$ ⓑ $\left(\frac{5}{4} , \frac{7}{4}\right)$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} 3 x + y = −3 \\ 2 x + 3 y = 5 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} −3 x + y = −1 \\ 2 x + y = 4 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} y = x + 2 \\ y = −2 x + 2 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} y = \frac{3}{2} x + 1 \\ y = - \frac{1}{2} x + 5 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} - x + y = −3 \\ 4 x + 4 y = 4 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} −3 x + y = −1 \\ 2 x + y = 4 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x + y = 5 \\ 2 x - y = 4 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x + y = 2 \\ x - y = 0 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x + y = −5 \\ x - y = 3 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x + y = −4 \\ - x + 2 y = −2 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} −2 x + 3 y = 3 \\ x + 3 y = 12 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} 2 x + 3 y = 6 \\ y = −2 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x - 3 y = −3 \\ y = 2 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} 2 x - y = −1 \\ x = 1 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x - 3 y = −6 \\ x = −3 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} −2 x + 4 y = 4 \\ y = \frac{1}{2} x \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x = −3 y + 4 \\ 2 x + 6 y = 8 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} 2 x + y = 6 \\ −8 x - 4 y = −24 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} x + 3 y = −6 \\ 4 y = - \frac{4}{3} x - 8 \end{cases}$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$\begin{cases} −3 x + 2 y = −2 \\ y = − x + 4 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = \frac{2}{3} x + 1 \\ −2 x + 3 y = 5 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = −2 x + 1 \\ 4 x + 2 y = 8 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = \frac{2}{3} x + 1 \\ 2 x - 3 y = 7 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} 4 x + 2 y = 10 \\ 4 x - 2 y = −6 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = - \frac{1}{2} x + 5 \\ x + 2 y = 10 \end{cases}$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$\begin{cases} y = 2 x + 3 \\ 2 x - y = −3 \end{cases}$

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?

Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

In a system of linear equations, the two equations have the same slope. Describe the possible solutions to the system.