Summary: Solving Equations Using the Division and Multiplication Properties of Equality

Key Concepts

• Division and Multiplication Properties of Equality
  • Division Property of Equality: For all real numbers a, b, c, and [latex]c\ne 0[/latex] , if [latex]a=b[/latex] , then [latex]ac=bc[/latex] .
  • Multiplication Property of Equality: For all real numbers a, b, c, if [latex]a=b[/latex] , then [latex]ac=bc[/latex] .

• Solve an equation with variables and constants on both sides
  1. Choose one side to be the variable side and then the other will be the constant side.
  2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
  3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable[latex]1[/latex], using the Multiplication or Division Property of Equality.
  5. Check the solution by substituting into the original equation.
• General strategy for solving linear equations
  1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
  2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
  3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable term equal to [latex]1[/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.
  5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

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