A46 APPENDIX Review Sometimes a linear equation is called a first-degree equation, because the left side is a polynomial in x of degree 1. Check: Let = − x 1 2 in the expression on the left side of the equation and simplify. Let = − x 1 2 in the expression on the right side of the equation and simplify. If the two expressions are equal, the solution checks. ( ) ( ) ( ) ( ) ( ) ( ) − = − − = − = − − = − − = − = − x x 3 2 3 1 2 2 3 5 2 15 2 5 1 5 1 2 1 5 3 2 15 2 Since the two expressions are equal, the solution = − x 1 2 checks. The solution set is { } − 1 2 . Solving a Linear Equation Solve the equation: ( ) ( ) − = − x x 3 2 5 1 Solution EXAMPLE 1 ( ) ( ) − = − − = − − − = − − − − = − − − + = − + − = − − = − = − x x x x x x x x x x x x x 3 2 5 1 3 6 5 5 3 6 5 5 5 5 2 6 5 2 6 6 5 6 2 1 2 2 1 2 1 2 Use the Distributive Property. Subtract x5 from both sides. Simplify. Add 6 to both sides. Simplify. Divide both sides by 2. − Simplify. Now Work problem 25 The next example illustrates the solution of an equation that does not appear to be linear but leads to a linear equation upon simplification. Solving an Equation That Leads to a Linear Equation Solve the equation: ( )( ) ( )( ) − − = − − x x x x 2 1 1 5 2 5 Solution EXAMPLE 2 ( )( ) ( )( ) − − = − − − + = − + − + − = − + − − + = − + − + − = − + − − = − + − + = − + + = = = x x x x x x x x x x x x x x x x x x x x x x x x x x x 2 1 1 5 2 5 2 3 1 2 15 25 2 3 1 2 2 15 25 2 3 1 15 25 3 1 1 15 25 1 3 15 24 3 15 15 24 15 12 24 12 12 24 12 2 2 2 2 2 2 2 Multiply and combine like terms. Subtract x2 2 from both sides. Simplify. Subtract 1 from both sides. Simplify. Add x 15 to both sides. Simplify. Divide both sides by 12. Simplify.
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