4.7 Graphing Systems of Linear Inequalities

Before you get started, take this readiness quiz.

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 $y > 2 x + 3 .$
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.

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

ⓐ $\left(5 , −2\right)$ ⓑ $\left(−1 , 3\right)$

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

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

ⓐ $\left(−4 , −1\right)$ ⓑ $(\text{8},\text{ 3})$

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

$\begin{cases} 6 x - 5 y < 20 \\ −2 x + 7 y > −8 \end{cases}$

ⓐ $(\text{1},\text{ }−\text{3})$ ⓑ $(−\text{4},\text{ 4})$

Graph the system $\begin{cases} x + 2 y \leq 6 \\ y \geq - \frac{1}{2} x - 4 \end{cases} .$ What does the solution mean?

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

$\begin{cases} x + y = 8 \\ y = x - 4 \end{cases}$
ⓐ $\left(6 , 2\right)$
ⓑ $\left(9 , −1\right)$

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

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

Solve the systems of equations by substitution.

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

Solve the systems of equations by substitution.

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

Solve the systems of equations by substitution.

$\begin{cases} y = - \frac{1}{3} x + 2 \\ x + 3 y = 6 \end{cases}$

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

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

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 and solve.

Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?

Translate to a system of equations and solve.

Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?

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.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

Translate to a system of equations and solve.

The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.

Translate to a system of equations and solve.

Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes $\left(\right. \frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

Translate to a system of equations and solve.

A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Translate to a system of equations and solve.

Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?

Translate to a system of equations and solve.

A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

Translate to a system of equations and solve.

When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?

Determine whether the ordered triple is a solution to the system.

$\begin{cases} y = \frac{2}{3} x - 2 \\ x + 3 y - z = 15 \\ x - 3 y + z = −2 \end{cases}$
ⓐ $\left(−6 , 5 , \frac{1}{2}\right)$
ⓑ $\left(5 , \frac{4}{3} , −3\right)$

Solve the system of equations.

$\begin{cases} x + \frac{5}{2} y + z = −2 \\ 2 x + 2 y + \frac{1}{2} z = −4 \\ \frac{1}{3} x - y - z = 1 \end{cases}$

Solve the system of equations.

$\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

Solve Applications using Systems of Linear Equations with Three Variables

After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is $135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends $115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their tot…

Write each system of linear equations as an augmented matrix.

$\begin{cases} 4 x + 3 y = −2 \\ x - 2 y - 3 z = 7 \\ 2 x - y + 2 z = −6 \end{cases}$

Write the system of equations that that corresponds to the augmented matrix.

$\left[\right. \begin{matrix}1 & & & 0 & & & −3 \\ 1 & & & −2 & & & 0 \\ 0 & & & −1 & & & 2\end{matrix} \left|\right. \begin{matrix}−1 \\ −2 \\ 3\end{matrix} \left]\right.$

Perform the indicated operations on the augmented matrices.

$\left[\right. \begin{aligned}1 & & & −3 & & & −2 \\ 2 & & & 2 & & & −1 \\ 4 & & & −2 & & & −3\end{aligned} \left|\right. \begin{matrix}4 \\ −3 \\ −1\end{matrix} \left]\right.$

ⓐ Interchange rows 2 and 3.
ⓑ Multiply row 1 by 2.
ⓒ Multiply row 3 by $−2$ and add to row 2.

Solve each system of equations using a matrix.

$\begin{cases} 2 x - y + 3 z = −3 \\ − x + 2 y - z = 10 \\ x + y + z = 5 \end{cases}$

Solve each system of equations using a matrix.

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

In the following exercise, evaluate the determinate of the square matrix.

$\begin{bmatrix} 8 & & & −4 \\ 5 & & & −3 \end{bmatrix}$

In the following exercise, evaluate each determinant by expanding by minors along the first row.

$\begin{vmatrix} −2 & & & −3 & & & −4 \\ 5 & & & −6 & & & 7 \\ −1 & & & 2 & & & 0 \end{vmatrix}$

Solve each system of equations using Cramer’s rule

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

Solve each system of equations using Cramer’s rule

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

Solve each system of equations using Cramer’s rule

$\begin{cases} 3 x + 4 y - 3 z = −2 \\ 2 x + 3 y - z = −12 \\ x + y - 2 z = 6 \end{cases}$

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

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

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

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

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

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

$\begin{cases} x + y - z = −1 \\ 2 x - y + 2 z = 8 \\ −3 x + 2 y + z = −9 \end{cases}$

Solve the system of equations using a matrix.

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

Solve using Cramer’s rule.

Evaluate the determinant by expanding by minors:
$\begin{vmatrix} 3 & & & −2 & & & −2 \\ 2 & & & −1 & & & 4 \\ −1 & & & 0 & & & −3 \end{vmatrix} .$

Translate to a system of equations and solve.

A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?

Translate to a system of equations and solve.

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a…