Chapter 5 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

coincident lines

y

complementary angles

Two angles are complementary if the sum of the measures of their angles is 90 degrees.

consistent system

A consistent system of equations is a system of equations with at least one solution.

dependent equations

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

inconsistent system

An inconsistent system of equations is a system of equations with no solution.

independent equations

Two equations are independent if they have different solutions.

solutions of a system of equations

x

y

supplementary angles

Two angles are supplementary if the sum of the measures of their angles is 180 degrees.

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations.

system of linear inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

##

Key Concepts

• To solve a system of linear equations by graphing Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Step 5. Check the solution in both equations.

• Determine the number of solutions from the graph of a linear system

• Determine the number of solutions of a linear system by looking at the slopes and intercepts

• Determine the number of solutions and how to classify a system of equations

• Problem Solving Strategy for Systems of Linear Equations Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose variables to represent those quantities. Step 4. Translate into a system of equations. Step 5. Solve the system of equations using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Solve a system of equations by substitution Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable. Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations.

• To Solve a System of Equations by Elimination Step 1. Write both equations in standard form. If any coefficients are fractions, clear them. Step 2. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations.

• Table for coin and mixture applications

• Table for concentration applications

• Table for interest applications

• To Solve a System of Linear Inequalities by Graphing Step 1. Graph the first inequality. Graph the boundary line. Shade in the side of the boundary line where the inequality is true. Step 2. On the same grid, graph the second inequality. Graph the boundary line. Shade in the side of that boundary line where the inequality is true. Step 3. The solution is the region where the shading overlaps. Step 4. Check by choosing a test point.

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Review Exercises

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.

Solve a System of Linear Equations by Graphing

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

Determine the Number of Solutions of a Linear System

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

Solve Applications of Systems of Equations by Graphing

Solve a System of Equations by Substitution

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

Solve Applications of Systems of Equations by Substitution

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

Solve a System of Equations by Elimination

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

Solve Applications of Systems of Equations by Elimination

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

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.

Translate to a System of Equations

In the following exercises, translate to a system of equations. Do not solve the system.

Solve Direct Translation Applications

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

Solve Geometry Applications

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

Solve Uniform Motion Applications

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

Solve Mixture Applications

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

Solve Interest Applications

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

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.

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Solve Applications of Systems of Inequalities

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

##

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

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

In the following exercises, solve each system of equations. Use either substitution or elimination.

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

Key Terms

coincident lines

y

complementary angles

Two angles are complementary if the sum of the measures of their angles is 90 degrees.

consistent system

A consistent system of equations is a system of equations with at least one solution.

dependent equations

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

inconsistent system

An inconsistent system of equations is a system of equations with no solution.

independent equations

Two equations are independent if they have different solutions.

solutions of a system of equations

x

y

supplementary angles

Two angles are supplementary if the sum of the measures of their angles is 180 degrees.

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations.

system of linear inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

##

Key Concepts

• To solve a system of linear equations by graphing Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Step 5. Check the solution in both equations.

• Determine the number of solutions from the graph of a linear system

• Determine the number of solutions of a linear system by looking at the slopes and intercepts

• Determine the number of solutions and how to classify a system of equations

• Problem Solving Strategy for Systems of Linear Equations Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose variables to represent those quantities. Step 4. Translate into a system of equations. Step 5. Solve the system of equations using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Solve a system of equations by substitution Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable. Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations.

• To Solve a System of Equations by Elimination Step 1. Write both equations in standard form. If any coefficients are fractions, clear them. Step 2. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations.

• Table for coin and mixture applications

• Table for concentration applications

• Table for interest applications

• To Solve a System of Linear Inequalities by Graphing Step 1. Graph the first inequality. Graph the boundary line. Shade in the side of the boundary line where the inequality is true. Step 2. On the same grid, graph the second inequality. Graph the boundary line. Shade in the side of that boundary line where the inequality is true. Step 3. The solution is the region where the shading overlaps. Step 4. Check by choosing a test point.

##

Review Exercises

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.

Solve a System of Linear Equations by Graphing

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

Determine the Number of Solutions of a Linear System

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

Solve Applications of Systems of Equations by Graphing

Solve a System of Equations by Substitution

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

Solve Applications of Systems of Equations by Substitution

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

Solve a System of Equations by Elimination

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

Solve Applications of Systems of Equations by Elimination

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

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.

Translate to a System of Equations

In the following exercises, translate to a system of equations. Do not solve the system.

Solve Direct Translation Applications

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

Solve Geometry Applications

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

Solve Uniform Motion Applications

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

Solve Mixture Applications

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

Solve Interest Applications

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

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.

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Solve Applications of Systems of Inequalities

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

##

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

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

In the following exercises, solve each system of equations. Use either substitution or elimination.

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

1. $\mathrm{\{}\begin{array}{l} x+3y=-9 \\ 2x-4y=12 \\ \end{array}$

ⓐ $(-3,-2)$ ⓑ $(0,-3)$

2. $\mathrm{\{}\begin{array}{l} 3x+y=6 \\ x+3y=-6 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 2x-y=6 \\ y=4 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 2x-y=5 \\ 4x-2y=10 \\ \end{array}$

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

3. $\mathrm{\{}\begin{array}{l} 5x-4y=0 \\ y=\frac{5}{4}x-5 \\ \end{array}$

1. 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?

2. $\mathrm{\{}\begin{array}{l} 3x-y=-5 \\ y=2x+4 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} x-y=0 \\ 2x+5y=-14 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y=-5x \\ 5x+y=6 \\ \end{array}$

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

3. 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.

1. $\mathrm{\{}\begin{array}{l} x+y=12 \\ x-y=-10 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 3x-8y=20 \\ x+3y=1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 9x+4y=2 \\ 5x+3y=5 \\ \end{array}$

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

2. $\mathrm{\{}\begin{array}{l} 6x-5y=27 \\ 3x+10y=-24 \\ \end{array}$

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

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

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

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

1. 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.

2. 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?

3. 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.

1. 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?

2. 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?

3. 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?

1. $\mathrm{\{}\begin{array}{l} 4x+y>6 \\ 3x-y\le{}12 \\ \end{array}$

ⓐ $(2,-1)$ ⓑ $(3,-2)$

2. $\mathrm{\{}\begin{array}{l} y<3x+1 \\ y\ge-x-2 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 2x-3y<6 \\ 3x+4y\ge{}12 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x+3y<5 \\ y\ge-\frac{1}{3}x+6 \\ \end{array}$

2. 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.

1. ⓐ Write a system of inequalities to model this situation.

2. ⓑ Graph the system.

3. ⓒ Should she display 26 bracelets and 14 necklaces?

4. ⓓ Should she display 39 bracelets and 1 necklace?

3. $\mathrm{\{}\begin{array}{l} x-4y=-8 \\ 2x+5y=10 \\ \end{array}$

ⓐ $(0,2)$ ⓑ $(4,3)$

1. $\mathrm{\{}\begin{array}{l} x-y>-2 \\ y\le{}3x+1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} x+y=-3 \\ x-y=11 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y=-\frac{4}{5}x+1 \\ 8x+10y=10 \\ \end{array}$

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

2. 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.

3. 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?

1. It takes $5\frac{1}{2}$ hours for a jet to fly 2,475 miles with a headwind from San Jose, California to Lihue, Hawaii. The return flight from Lihue to San Jose with a tailwind, takes 5 hours. Find the speed of the jet in still air and the speed of the wind.

2. 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?

3. Translate to a system of inequalities and solve.

Andi wants to spend no more than $50 on Halloween treats. She wants to buy candy bars that cost$1 each and lollipops that cost $0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.

1. ⓐ Write a system of inequalities to model this situation.

2. ⓑ Graph the system.

3. ⓒ Can she buy 20 candy bars and 70 lollipops?

4. ⓓ Can she buy 15 candy bars and 65 lollipops?