5.6 Graphing Systems of Linear Inequalities

Solve the inequality $2 a < 5 a + 12$.
If you missed this problem, review the reference.

Determine whether the ordered pair $\left(3 , \frac{1}{2}\right)$ is a solution to the system $\begin{cases} x + 2 y = 4 \\ y = 6 x \end{cases}$.
If you missed this problem, review the reference

Determine whether the ordered pair is a solution to the system.
$\begin{cases} x - 5 y > 10 \\ 2 x + 3 y > −2 \end{cases}$

ⓐ $\left(3 , −1\right)$ ⓑ $\left(6 , −3\right)$

Determine whether the ordered pair is a solution to the system.
$\begin{cases} y > 4 x - 2 \\ 4 x - y < 20 \end{cases}$

ⓐ $\left(2 , 1\right)$ ⓑ $\left(4 , −1\right)$

Determine whether each ordered pair is a solution to the system.

$\begin{cases} 3 x + y > 5 \\ 2 x - y \leq 10 \end{cases}$

ⓐ $\left(3 , −3\right)$ ⓑ $\left(7 , 1\right)$

Determine whether each ordered pair is a solution to the system.

$\begin{cases} y > \frac{2}{3} x - 5 \\ x + \frac{1}{2} y \leq 4 \end{cases}$

ⓐ $\left(6 , −4\right)$ ⓑ $\left(3 , 0\right)$

Determine whether each ordered pair is a solution to the system.

$\begin{cases} 7 x + 2 y > 14 \\ 5 x - y \leq 8 \end{cases}$

ⓐ $\left(2 , 3\right)$ ⓑ $\left(7 , −1\right)$

Determine whether each ordered pair is a solution to the system.

$\begin{cases} 2 x + 3 y \geq 2 \\ 4 x - 6 y < −1 \end{cases}$

ⓐ $\left(\frac{3}{2} , \frac{4}{3}\right)$ ⓑ $\left(\frac{1}{4} , \frac{7}{6}\right)$

Graph the inequality $x - y \geq 3$. How do you know which side of the line $x - y = 3$ should be shaded?

Determine if the following points are solutions to the given system of equations.

$\begin{cases} x + 3 y = −9 \\ 2 x - 4 y = 12 \end{cases}$

ⓐ $\left(−3 , −2\right)$ ⓑ $\left(0 , −3\right)$

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

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

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

$\begin{cases} 5 x - 4 y = 0 \\ y = \frac{5}{4} x - 5 \end{cases}$

Solve Applications of Systems of Equations by Graphing

LaVelle is making a pitcher of caffe mocha. For each ounce of chocolate syrup, she uses five ounces of coffee. How many ounces of chocolate syrup and how many ounces of coffee does she need to make 48 ounces of caffe mocha?

Solve the systems of equations by substitution.

$\begin{cases} 3 x - y = −5 \\ y = 2 x + 4 \end{cases}$

Solve the systems of equations by substitution.

$\begin{cases} x - y = 0 \\ 2 x + 5 y = −14 \end{cases}$

Solve the systems of equations by substitution.

$\begin{cases} y = −5 x \\ 5 x + y = 6 \end{cases}$

Translate to a system of equations and solve.

The sum of two number is 55. One number is 11 less than the other. Find the numbers.

Translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 2 less than 3 times the measure of the other small angle. Find the measure of both angles.

Solve the systems of equations by elimination.

$\begin{cases} x + y = 12 \\ x - y = −10 \end{cases}$

Solve the systems of equations by elimination.

$\begin{cases} 3 x - 8 y = 20 \\ x + 3 y = 1 \end{cases}$

Solve the systems of equations by elimination.

$\begin{cases} 9 x + 4 y = 2 \\ 5 x + 3 y = 5 \end{cases}$

Translate to a system of equations and solve.

The sum of two numbers is $−90$. Their difference is $16$. Find the numbers.

Decide whether it would be more convenient to solve the system of equations by substitution or elimination.

$\begin{cases} 6 x - 5 y = 27 \\ 3 x + 10 y = −24 \end{cases}$

Translate to a system of equations. Do not solve the system.

The sum of two numbers is $−32$. One number is two less than twice the other. Find the numbers.

Translate to a system of equations. Do not solve the system.

Last month Jim and Debbie earned $7,200. Debbie earned $1,600 more than Jim earned. How much did they each earn?

Translate to a system of equations and solve.

Pam is 3 years older than her sister, Jan. The sum of their ages is 99. Find their ages.

Translate to a system of equations and solve.

The difference of two supplementary angles is 58 degrees. Find the measures of the angles.

Translate to a system of equations and solve.

Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.

Translate to a system of equations and solve.

Sheila and Lenore were driving to their grandmother’s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?

Translate to a system of equations and solve.

Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.

Translate to a system of equations and solve.

Lynn paid a total of $2,780 for 261 tickets to the theater. Student tickets cost $10 and adult tickets cost $15. How many student tickets and how many adult tickets did Lynn buy?

Translate to a system of equations and solve.

Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her $1.29 per cup. The candies are $2.49 per cup and the nuts are $0.69 per cup. How many cups of candies and how many cups of nuts should she use?

Translate to a system of equations and solve.

Jack has $12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?

Determine whether each ordered pair is a solution to the system.

$\begin{cases} 4 x + y > 6 \\ 3 x - y \leq 12 \end{cases}$

ⓐ $\left(2 , −1\right)$ ⓑ $\left(3 , −2\right)$

For the following set of equations, determine if each ordered pair is a solution.

$\begin{cases} x - 4 y = −8 \\ 2 x + 5 y = 10 \end{cases}$

ⓐ $\left(0 , 2\right)$ ⓑ $\left(4 , 3\right)$

Solve each system of equations. Use either substitution or elimination.

$\begin{cases} x + y = −3 \\ x - y = 11 \end{cases}$

Solve each system of equations. Use either substitution or elimination.

$\begin{cases} y = - \frac{4}{5} x + 1 \\ 8 x + 10 y = 10 \end{cases}$

Translate to a system of equations and solve.

The sum of two numbers is −24. One number is 104 less than the other. Find the numbers.

Translate to a system of equations and solve.

Two angles are complementary. The measure of the larger angle is six more than twice the measure of the smaller angle. Find the measures of both angles.

Translate to a system of equations and solve.

Kathy left home to walk to the mall, walking quickly at a rate of 4 miles per hour. Her sister Abby left home 15 minutes later and rode her bike to the mall at a rate of 10 miles per hour. How long will it take Abby to catch up to Kathy?

Translate to a system of equations and solve.

Liz paid $160 for 28 tickets to take the Brownie troop to the science museum. Children’s tickets cost $5 and adult tickets cost $9. How many children’s tickets and how many adult tickets did Liz buy?