8.S: Solving Linear Equations (Summary)
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
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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.
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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.
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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.
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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
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Division and Multiplication Properties of Equality
- Division Property of Equality: For all real numbers a, b, c, and c ≠ 0, if a = b, then \(\dfrac{a}{c} = \dfrac{b}{c}\).
- 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
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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.
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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
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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 and Attributions
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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 ."