### Review Exercises

## Solve Systems of Linear Equations with Two Variables

**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.

$\{\begin{array}{c}x+3y=\mathrm{-9}\hfill \\ 2x-4y=12\hfill \end{array}$

ⓐ $(\mathrm{-3},\mathrm{-2})$

ⓑ $(0,\mathrm{-3})$

$\{\begin{array}{c}x+y=8\hfill \\ y=x-4\hfill \end{array}$

ⓐ $\left(6,2\right)$

ⓑ $\left(9,\mathrm{-1}\right)$

**Solve a System of Linear Equations by Graphing**

In the following exercises, solve the following systems of equations by graphing.

$\{\begin{array}{c}3x+y=6\hfill \\ x+3y=\mathrm{-6}\hfill \end{array}$

$\{\begin{array}{c}2x-y=5\hfill \\ 4x-2y=10\hfill \end{array}$

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

$\{\begin{array}{c}y=\frac{2}{5}x+2\hfill \\ \mathrm{-2}x+5y=10\hfill \end{array}$

$\{\begin{array}{c}5x-4y=0\hfill \\ y=\frac{5}{4}x-5\hfill \end{array}$

**Solve a System of Equations by Substitution**

In the following exercises, solve the systems of equations by substitution.

$\{\begin{array}{c}x-y=0\hfill \\ 2x+5y=\mathrm{-14}\hfill \end{array}$

$\{\begin{array}{c}y=\mathrm{-5}x\hfill \\ 5x+y=6\hfill \end{array}$

**Solve a System of Equations by Elimination**

In the following exercises, solve the systems of equations by elimination

$\{\begin{array}{c}x+y=12\hfill \\ x-y=\mathrm{-10}\hfill \end{array}$

$\{\begin{array}{c}9x+4y=2\hfill \\ 5x+3y=5\hfill \end{array}$

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

$\{\begin{array}{c}\text{\u2212}x+3y=8\hfill \\ 2x-6y=\mathrm{-20}\hfill \end{array}$

**Choose the Most Convenient Method to Solve a System of Linear Equations**

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

$\{\begin{array}{c}y=3x-9\hfill \\ 4x-5y=23\hfill \end{array}$

## Solve Applications with Systems of Equations

**Solve Direct Translation Applications**

In the following exercises, 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?

Ashanti has been offered positions by two phone companies. The first company pays a salary of $22,000 plus a commission of $100 for each contract sold. The second pays a salary of $28,000 plus a commission of $25 for each contract sold. How many contract would need to be sold to make the total pay the same?

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?

Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for $116. Lisa bought two notebooks and three thumb drives for $68. Find the cost of each notebook and each thumb drive.

**Solve Geometry Applications**

In the following exercises, translate to a system of equations and solve.

Two angles are complementary. The measure of the larger angle is five more than four times the measure of the smaller angle. Find the measures of both angles.

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.

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.

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.

**Solve Uniform Motion Applications**

In the following exercises, 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?

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 $(\frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

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.

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.

## Solve Mixture Applications with Systems of Equations

**Solve Mixture Applications with Systems of Equations**

For the following exercises, 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?

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?

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?

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?

**Solve Interest Applications**

For the following exercises, 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?

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?

## Solve Systems of Equations with Three Variables

**Solve Systems of Equations with Three Variables**

In the following exercises, determine whether the ordered triple is a solution to the system.

$\{\begin{array}{c}3x-4y-3z=2\hfill \\ 2x-6y+z=3\hfill \\ 2x+3y-2z=3\hfill \end{array}$

ⓐ $(2,3,\mathrm{-1})$

ⓑ $(3,1,3)$

$\{\begin{array}{c}y=\frac{2}{3}x-2\hfill \\ x+3y-z=15\hfill \\ x-3y+z=\mathrm{-2}\hfill \end{array}$

ⓐ $\left(\mathrm{-6},5,\frac{1}{2}\right)$

ⓑ $\left(5,\frac{4}{3},\mathrm{-3}\right)$

**Solve a System of Linear Equations with Three Variables**

In the following exercises, solve the system of equations.

$\{\begin{array}{c}3x-5y+4z=5\hfill \\ 5x+2y+z=0\hfill \\ 2x+3y-2z=3\hfill \end{array}$

$\{\begin{array}{c}x+\frac{5}{2}y+z=\mathrm{-2}\hfill \\ 2x+2y+\frac{1}{2}z=\mathrm{-4}\hfill \\ \frac{1}{3}x-y-z=1\hfill \end{array}$

$\{\begin{array}{c}5x+3y=\mathrm{-6}\hfill \\ 2y+3z=\mathrm{-1}\hfill \\ 7x+z=1\hfill \end{array}$

$\{\begin{array}{c}\text{\u2212}x-3y+2z=14\hfill \\ \text{\u2212}x+2y-3z=\mathrm{-4}\hfill \\ 3x+y-2z=6\hfill \end{array}$

**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 total is $85. What is the cost of each item?

## Solve Systems of Equations Using Matrices

**Write the Augmented Matrix for a System of Equations.**

Write each system of linear equations as an augmented matrix.

$\{\begin{array}{c}3x-y=\mathrm{-1}\hfill \\ \mathrm{-2}x+2y=5\hfill \end{array}$

$\{\begin{array}{c}4x+3y=\mathrm{-2}\hfill \\ x-2y-3z=7\hfill \\ 2x-y+2z=\mathrm{-6}\hfill \end{array}$

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

$[\begin{array}{cccc}2\hfill & & & \hfill \mathrm{-4}\\ 3\hfill & & & \hfill \mathrm{-3}\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill \mathrm{-2}\\ \hfill \mathrm{-1}\end{array}]$

$[\begin{array}{ccccccc}\hfill 1& & & \hfill 0& & & \hfill \mathrm{-3}\\ \hfill 1& & & \hfill \mathrm{-2}& & & \hfill 0\\ \hfill 0& & & \hfill \mathrm{-1}& & & \hfill 2\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill \mathrm{-1}\\ \hfill \mathrm{-2}\\ \hfill 3\end{array}]$

In the following exercises, perform the indicated operations on the augmented matrices.

$[\begin{array}{cccc}4\hfill & & & \hfill \mathrm{-6}\\ 3\hfill & & & \hfill 2\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill \mathrm{-3}\\ \hfill 1\end{array}]$

ⓐ Interchange rows 2 and 1.

ⓑ Multiply row 1 by 4.

ⓒ Multiply row 2 by 3 and add to row 1.

$[\begin{array}{ccccccc}1\hfill & & & \hfill \mathrm{-3}& & & \hfill \mathrm{-2}\\ 2\hfill & & & \hfill 2& & & \hfill \mathrm{-1}\\ 4\hfill & & & \hfill \mathrm{-2}& & & \hfill \mathrm{-3}\end{array}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill 4\\ \hfill \mathrm{-3}\\ \hfill \mathrm{-1}\end{array}]$

ⓐ Interchange rows 2 and 3.

ⓑ Multiply row 1 by 2.

ⓒ Multiply row 3 by $\mathrm{-2}$ and add to row 2.

**Solve Systems of Equations Using Matrices**

In the following exercises, solve each system of equations using a matrix.

$\{\begin{array}{c}4x+y=6\hfill \\ x-y=4\hfill \end{array}$

$\{\begin{array}{c}2x-y+3z=\mathrm{-3}\hfill \\ \text{\u2212}x+2y-z=10\hfill \\ x+y+z=5\hfill \end{array}$

$\{\begin{array}{c}2y+3z=\mathrm{-1}\hfill \\ 5x+3y=\mathrm{-6}\hfill \\ 7x+z=1\hfill \end{array}$

$\{\begin{array}{c}x+y-3z=\mathrm{-1}\hfill \\ y-z=0\hfill \\ \text{\u2212}x+2y=1\hfill \end{array}$

## Solve Systems of Equations Using Determinants

**Evaluate the Determinant of a 2 × 2 Matrix**

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

$\left[\begin{array}{cccc}8\hfill & & & \hfill \mathrm{-4}\\ 5\hfill & & & \hfill \mathrm{-3}\end{array}\right]$

**Evaluate the Determinant of a 3 × 3 Matrix**

In the following exercise, find and then evaluate the indicated minors.

$\left|\begin{array}{ccccccc}\hfill \mathrm{-1}& & & \hfill \mathrm{-3}& & & \hfill 2\\ \hfill 4& & & \hfill \mathrm{-2}& & & \hfill \mathrm{-1}\\ \hfill \mathrm{-2}& & & \hfill 0& & & \hfill \mathrm{-3}\end{array}\right|;$ Find the minor ⓐ ${a}_{1}$ ⓑ ${b}_{1}$ ⓒ ${c}_{2}$

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

$\left|\begin{array}{ccccccc}\hfill \mathrm{-2}& & & \hfill \mathrm{-3}& & & \hfill \mathrm{-4}\\ \hfill 5& & & \hfill \mathrm{-6}& & & \hfill 7\\ \hfill \mathrm{-1}& & & \hfill 2& & & \hfill 0\end{array}\right|$

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

$\left|\begin{array}{ccccccc}\hfill 3& & & \hfill 5& & & \hfill 4\\ \hfill \mathrm{-1}& & & \hfill 3& & & \hfill 0\\ \hfill \mathrm{-2}& & & \hfill 6& & & \hfill 1\end{array}\right|$

**Use Cramer’s Rule to Solve Systems of Equations**

In the following exercises, solve each system of equations using Cramer’s rule

$\{\begin{array}{c}4x-3y+z=7\hfill \\ 2x-5y-4z=3\hfill \\ 3x-2y-2z=\mathrm{-7}\hfill \end{array}$

$\{\begin{array}{c}x+y-3z=\mathrm{-1}\hfill \\ y-z=0\hfill \\ \text{\u2212}x+2y=1\hfill \end{array}$

$\{\begin{array}{c}3x+4y-3z=\mathrm{-2}\hfill \\ 2x+3y-z=\mathrm{-12}\hfill \\ x+y-2z=6\hfill \end{array}$

**Solve Applications Using Determinants**

In the following exercises, determine whether the given points are collinear.

$\left(0,2\right),$ $\left(\mathrm{-1},\mathrm{-1}\right),$ and $\left(\mathrm{-2},4\right)$

## Graphing Systems of Linear Inequalities

**Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities**

In the following exercises, determine whether each ordered pair is a solution to the system.

$\{\begin{array}{c}4x+y>6\hfill \\ 3x-y\le 12\hfill \end{array}$

ⓐ $(2,\mathrm{-1})$

ⓑ $(3,\mathrm{-2})$

$\{\begin{array}{c}y>\frac{1}{3}x+2\hfill \\ x-\frac{1}{4}y\le 10\hfill \end{array}$

ⓐ $(6,5)$

ⓑ $(15,8)$

**Solve a System of Linear Inequalities by Graphing**

In the following exercises, solve each system by graphing.

$\{\begin{array}{c}x-y>\mathrm{-1}\hfill \\ y<\frac{1}{3}x-2\hfill \end{array}$

$\{\begin{array}{c}y\le -\frac{3}{4}x+1\hfill \\ x\ge \mathrm{-5}\hfill \end{array}$

$\{\begin{array}{c}y\ge 2x-5\hfill \\ \mathrm{-6}x+3y>\mathrm{-4}\hfill \end{array}$

**Solve Applications of Systems of Inequalities**

In the following exercises, translate to a system of inequalities and solve.

Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for $12 each and the necklaces for $18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least $500 worth in order to earn a profit.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓒ Should she display 26 bracelets and 14 necklaces?

ⓓ Should she display 39 bracelets and 1 necklace?

Annie has a budget of $600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost $4 each and hardcover books cost $15 each.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓒ Can she buy 8 paperback books and 40 hardcover books?

ⓓ Can she buy 10 paperback books and 37 hardcover books?