# 8.S: Solving Linear Equations (Summary)

- Page ID
- 5023

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## Key Terms

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

## Key Concepts

### 8.1 - Solve Equations Using the Subtraction and Addition Properties of Equality

**Determine whether a number is a solution to an equation.**- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- 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 Equality**- Subtraction Property of Equality: For all real numbers a, b, and c, if a = b then a - c = b - c.
- Addition Property of Equality: For all real numbers a, b, and c, if a = b then a + c = b + c.

**Translate a word sentence to an algebraic equation.**- Locate the “equals” word(s). Translate to an equal sign.
- Translate the words to the left of the “equals” word(s) into an algebraic expression.
- Translate the words to the right of the “equals” word(s) into an algebraic expression.

**Problem-solving strategy**- Read the problem. Make sure you understand all the words and ideas.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- 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.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.

### 8.2 - Solve Equations Using the Division and Multiplication Properties of Equality

**Division and Multiplication Properties of Equality**- Division Property of Equality: For all real numbers a, b, c, and c ≠ 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.

### 8.3 - Solve Equations with Variables and Constants on Both Sides

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

**General strategy for solving linear equations**- Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
- Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
- Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
- 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.
- Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

### 8.4 - Solve Equations with Fraction or Decimal Coefficients

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

## Contributors

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."