Chapter 8 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

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

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Key Concepts

• Determine whether a number is a solution to an equation.Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Subtraction and Addition Properties of EqualitySubtraction Property of EqualityFor all real numbers a, b, and c,if a = b then $a-c=b-c$. Addition Property of EqualityFor all real numbers a, b, and c,if a = b then $a+c=b+c$.

• Translate a word sentence to an algebraic equation.Step 1. Locate the “equals” word(s). Translate to an equal sign. Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

• Problem-solving strategyStep 1. Read the problem. Make sure you understand all the words and ideas. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation 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.

• Division and Multiplication Properties of EqualityDivision Property of Equality: For all real numbers a, b, c, and $c\ne{}0$, if $a=b$, then $\frac{a}{c}=\frac{b}{c}$. Multiplication Property of Equality: For all real numbers a, b, c, if $a=b$, then $ac=bc$.

• Solve an equation with variables and constants on both sidesStep 1. Choose one side to be the variable side and then the other will be the constant side. Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality. Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality. Step 5. Check the solution by substituting into the original equation.

• General strategy for solving linear equationsStep 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms. Step 2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality. Step 3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation. Step 5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

• Solve equations with fraction coefficients by clearing the fractions.Step 1. Find the least common denominator of all the fractions in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions. Step 3. Solve using the General Strategy for Solving Linear Equations.

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Review Exercises

In the following exercises, determine whether the given number is a solution to the equation.

In the following exercises, solve the equation using the Subtraction Property of Equality.

In the following exercises, solve the equation using the Addition Property of Equality.

In the following exercises, solve the equation.

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

In the following exercises, translate into an algebraic equation and solve.

In the following exercises, solve each equation using the Division Property of Equality.

In the following exercises, solve each equation using the Multiplication Property of Equality.

In the following exercises, solve each equation.

In the following exercises, solve the equations with constants on both sides.

In the following exercises, solve the equations with variables on both sides.

In the following exercises, solve the equations with constants and variables on both sides.

In the following exercises, solve each linear equation using the general strategy.

In the following exercises, solve each equation by clearing the fractions.

In the following exercises, solve each equation by clearing the decimals.

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In the following exercises, solve each equation.

Key Terms

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

##

Key Concepts

• Determine whether a number is a solution to an equation.Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Subtraction and Addition Properties of EqualitySubtraction Property of EqualityFor all real numbers a, b, and c,if a = b then $a-c=b-c$. Addition Property of EqualityFor all real numbers a, b, and c,if a = b then $a+c=b+c$.

• Translate a word sentence to an algebraic equation.Step 1. Locate the “equals” word(s). Translate to an equal sign. Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

• Problem-solving strategyStep 1. Read the problem. Make sure you understand all the words and ideas. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation 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.

• Division and Multiplication Properties of EqualityDivision Property of Equality: For all real numbers a, b, c, and $c\ne{}0$, if $a=b$, then $\frac{a}{c}=\frac{b}{c}$. Multiplication Property of Equality: For all real numbers a, b, c, if $a=b$, then $ac=bc$.

• Solve an equation with variables and constants on both sidesStep 1. Choose one side to be the variable side and then the other will be the constant side. Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality. Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality. Step 5. Check the solution by substituting into the original equation.

• General strategy for solving linear equationsStep 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms. Step 2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality. Step 3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation. Step 5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

• Solve equations with fraction coefficients by clearing the fractions.Step 1. Find the least common denominator of all the fractions in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions. Step 3. Solve using the General Strategy for Solving Linear Equations.

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Review Exercises

In the following exercises, determine whether the given number is a solution to the equation.

In the following exercises, solve the equation using the Subtraction Property of Equality.

In the following exercises, solve the equation using the Addition Property of Equality.

In the following exercises, solve the equation.

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

In the following exercises, translate into an algebraic equation and solve.

In the following exercises, solve each equation using the Division Property of Equality.

In the following exercises, solve each equation using the Multiplication Property of Equality.

In the following exercises, solve each equation.

In the following exercises, solve the equations with constants on both sides.

In the following exercises, solve the equations with variables on both sides.

In the following exercises, solve the equations with constants and variables on both sides.

In the following exercises, solve each linear equation using the general strategy.

In the following exercises, solve each equation by clearing the fractions.

In the following exercises, solve each equation by clearing the decimals.

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In the following exercises, solve each equation.

1. $x+16=31,\,x=15$

2. $-9n=45,\,n=54$

3. $a+\frac{1}{3}=\frac{5}{3}$

1. $u-7=10$

2. $c-\frac{3}{11}=\frac{9}{11}$

3. $n-12=32$

1. $f+\frac{2}{3}=4$

2. $y+8-15=-3$

3. $7x+10-6x+3=5$

1. $6(n-1)-5n=-14$

2. The sum of $-6$ and $m$ is $25.$

3. Rochelle’s daughter is $11$ years old. Her son is $3$ years younger. How old is her son?

1. Peter paid $\text{\textdollar 9.75}$ to go to the movies, which was $\text{\textdollar 46.25}$ less than he paid to go to a concert. How much did he pay for the concert?

2. $8x=72$

3. $0.25p=5.25$

1. $\frac{n}{6}=18$

2. $36=\frac{3}{4}\,x$

3. $-18m=-72$

1. $0.45x=6.75$

2. $5r-3r+9r=35-2$

3. $8p+7=47$

1. $3x+19=-47$

2. $7y=6y-13$

3. $k=-6k-35$

1. $12x-9=3x+45$

2. $4u+16=-19-u$

3. $6(x+6)=24$

1. $-(s+4)=18$

2. $23-3(y-7)=8$

3. $8(r-2)=6(r+10)$

1. $4(3.5y+0.25)=365$

2. $\frac{2}{5}\,n-\frac{1}{10}=\frac{7}{10}$

3. $\frac{3}{4}\,a-\frac{1}{3}=\frac{1}{2}\,a+\frac{5}{6}$

1. $0.8x-0.3=0.7x+0.2$

2. $0.6p-1.9=0.78p+1.7$

3. Determine whether each number is a solution to the equation. $3x+5=23.$

1. ⓐ $\,6$

2. ⓑ $\,\frac{23}{5}$

1. $9c=144$

2. $-8x-15+9x-1=-21$

3. $\frac{2}{3}\,x=6$

1. $10y=-5y+60$

2. $9m-2-4m+m=42-8$

3. $-(d+9)=23$

1. $2(6x+5)-8=-22$

2. $\frac{1}{4}\,p+\frac{1}{3}=\frac{1}{2}$

3. Translate and solve: The difference of twice $x$ and $4$ is $16.$