300 CHAPTER 6 Algebra, Graphs, and Functions The division property of equality indicates that both sides of an equation can be divided by the same nonzero number without changing the solution. Note that the divisor, c, cannot be 0 because division by 0 is undefined. Division Property of Equality If = a b, then = a c b c for all real numbers a b , , and c, ≠ c 0. Example 7 Using the Division Property of Equality Solve the equation x9 54. − = − Solution To solve this equation, we isolate the variable by dividing both sides of the equation by −9. − = − − − = − − = x x x 9 54 9 9 54 9 6 7 Now try Exercise 51 An algorithm is a general procedure for accomplishing a task. The following general procedure is an algorithm for solving linear (or first-degree) equations. Sometimes the solution to an equation may be determined more easily by using a variation of this general procedure. Remember that the primary objective in solving any equation is to isolate the variable. A GENERAL PROCEDURE FOR SOLVING LINEAR EQUATIONS 1. If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple) of the fractions in the equation. This step will eliminate all fractions from the equation. 2. Use the distributive property to remove parentheses when necessary. 3. Combine like terms on the same side of the equal sign when possible. 4. Use the addition or subtraction property to collect all terms with a variable on one side of the equation and all constants on the other side of the equation. It may be necessary to use the addition or subtraction property more than once. This process will eventually result in an equation of the form = ax b, where a and b are real numbers. 5. Solve for the variable using the division or multiplication property. The result will be an answer in the form = x c, where c is a real number. 6. Check your answer by substituting the value obtained in Step 5 back into the original equation. PROCEDURE Example 8 Solving a Linear Equation Solve the equation − = x3 5 28. Solution Our goal is to isolate the variable; therefore we start by getting the term x3 by itself on one side of the equation. − = − + = + = x x x 3 5 28 3 5 5 28 5 3 33 Add 5 to both sides of the equation (addition property) (Step 4).
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