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Transforming Equations:β¨Multiplication and Division September 22, 2011 Transforming Equations Objective To solve equations using multiplication or division. Transforming Equations At a hardware store, small construction supplies are often sold by the pound rather than by the number of items. Suppose a pound of roof nails costs the same as a pound of floor nails. You would expect to pay the same price for two pounds of roof nails as for two pounds of floor nails, and the same price for one-half pound of roof nails as for one-half pound of floor nails. Multiplication Property of Equality If a, b, and c are any real numbers, and π = π, then ππ = ππ and ππ = ππ If equal numbers are multiplied by the same number, the products are equal. Division Property of Equality If a and b are any real numbers , c is any nonzero real number, and π = π, then π π = π π If equal numbers are divided by the same nonzero number, the quotients are equal. Transforming an Equation into an Equivalent Equation These properties give you two more ways to transform an equation into anβ¨equivalent equation. Transforming an Equation into an Equivalent Equation Transformation by Multiplication: Multiply each side of a given equation by the same nonzero real number. Transformation by Division: Divide each side of a given equation by the same nonzero real number. Example 1 Solve 6π₯ = 222. Solution 6π₯ = 222 Copy the equation. 6π₯ 222 = 6 6 Divide to each side by 6. π₯ = 37 Simplify. Example 1 Solve 6π₯ = 222. Check 6π₯ = 222 Copy the equation. 6 37 = 222 Substitute 37 for x. 222 = 222 ο The solution set is {37}. Example 2 Solve 8= 2 β π‘. 3 Solution 2 8=β π‘ 3 3 3 2 β 8 =β β π‘ 2 2 3 β12 = π‘ Copy the equation. To get t alone on one side, multiply each 3 side by β , the 2 reciprocal of β Simplify. 2 3 Example 2 Solve 8= 2 β π‘. 3 Check 2 8=β π‘ 3 Copy the equation. 2 8 = β β12 3 Substitute ο12 for t. 8=8 ο The solution set is {ο12}. Equivalent Equations You know that zero cannot be a divisor. Do you know why zero is not allowed as a multiplier in transforming an equation? Look at the following equations. 1. 2. 3. 4. 5π§ = 45 0 β 5π§ = 0 β 45 0 β 5 π§ = 0 β 45 0βπ§ =0 Equivalent Equations Equation (1) had just one root, namely 9. Equation (4) is satisfied by any real number. Since they do not have the same solution set, Equations (1) and (4)β¨are not equivalent. When transforming an equation, never multiply by zero! Class work Oral Exercises p 104: 1-18 Homework p 104: 3-39 mult of 3, p 105: prob 7-15 odd, p 106 MR