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Lesson 2 – 4: Solving Equations with Variables on Both Sides 2-4.1 - Solving equations with variables on both sides To solve equation that have variables on both sides we must collect the variables terms so that they are on only one side. We do this be adding or subtracting the variables to make one side zero. This is step 3 of the 5 for solving equations 5 Steps to solve Equations: Distribute to get rid of Parentheses Combine like Terms Get Variables to one side Inverse of Addition and Subtraction Inverse of Multiplication and Division Examples: Solve each equation 5 x 2 3x 4 7 k 4k 15 4k 4k 3k 15 k 5 3 x 3x 2x 2 4 2 2x 6 2 x3 2-4.2 - Simplifying both sides before solving Examples: Solve each equation 2( y 6) 3 y 2 y 12 3 y 2 y 2y 12 y 3 5 x 2 x 2 2(1 x) 3 5 x 2 x 2 2 2 x 3 3x 4 2 x 3x 3x 3 4 5 x 4 4 7 5x 7 x 5 2-4.3 - Equations with infinitely many solutions or no solutions Vocabulary: Identity - an equation that is true for all values of the variable. It has infinite solutions Contradiction - an equation that is not true for any value of the variable. It has no solutions Sometimes equations are true no matter what value is substituted in for the variable. In this case, the equation is called an identity and is said to have infinitely many solutions. On the other hand, sometimes no matter what value is put into an equation for the variable nothing will work and it has no solutions. This is said to be a contradiction. Examples: Solve each equation x 4 6 x 6 5x 2 5 x 4 4 5 x 5 x 5x 44 Infinite number of solutions 8 x 6 9 x 17 x x 6 17 x x x 6 17 No solutions