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Transforming Equations: Addition and Subtraction September 21, 2011 Transforming Equations Objective To solve equations using addition and subtraction. Transforming Equations Two soccer teams are tied at half time: 2 to 2. If each team scores 3 goals in the second half, the score will still be tied: 2+3=2+3 Two sporting goods stores charge $36 for a soccer ball. If, during a spring sale, each store reduces the price by $5, both stores will still be charging the same price: 36 β 5 = 36 β 5 Addition Property of Equality If a, b, and c are any real numbers, and π = π, then π + π = π + π and π + π = π + π If the same number is added to equal numbers, the sums are equal. Subtraction Property of Equality If a, b, and c are any real numbers, and π = π, then πβπ=πβπ If the same number is subtracted from equal numbers, the differences are equal. Subtraction Property of Equality The subtraction property is just a special case of the addition property of equality, since subtracting the number c is the same as adding οc. The addition property of equality guarantees that if π = π, then π + βπ = π + βπ or πβπ =πβπ Solving Equations The following examples show how to use the addition and subtraction properties of equality to solve some equations. You add the same number to, or subtract the same number from, each side of the equation in order to get an equation with the variable alone on one side of the equation. Example 1 Solve π₯ β 8 = 17. Solution π₯ β 8 = 17 Copy the equation. π₯ β 8 + 8 = 17 + 8 Add 8 to each side. π₯ = 25 Simplify. Example 1 Solve π₯ β 8 = 17. Check π₯ β 8 = 17 Copy the equation. 25 β 8 = 17 Substitute 25 for x. 17 = 17 ο The solution set is {25}. The properties of real numbers guarantee in Example 1 that if the original equation, π₯ β 8 = 17, is true for some value of x, then the final equation, π₯ = 25, is also true for that value of x, and vice versa. Therefore, the two equations have the same solution set, {25}. Example 2 Solve β5 = π + 13. Solution β5 = π + 13 Copy the equation. β5 β 13 = π + 13 β 13 Subtract 13 from each side. β18 = π Simplify. Example 2 Solve β5 = π + 13. Check β5 = π + 13 Copy the equation. β5 = β18 + 13 Substitute ο18 for n. β5 = β5 ο The solution set is {ο18}. Equivalent Equations Equations having the same solution set over a givenβ¨domain are called equivalent equations. To solve an equation you usuallyβ¨change, or transform, it into a simple equivalent equation whose solution set isβ¨easy to see. Solving Equations To solve an equation, transform it to an equation that has the variable isolated on one side of the equal sign. Transform, π₯ + 12 = 47 into π₯ = 35. Therefore, the solution set is, {35}. Transformations That Produce Equivalent Equations 1. Transforming by Substitution Substitute an equivalent expression for any expression in the given equation. 2. Transformation by Addition Add the same real number to each side of the given equation. 3. Transformation by Subtraction Subtract the same real number from each side of the given equation. Class work Oral Exercises p 97: 1-21 Homework p 97: 1-39 odd, p 98: prob. 1-9 odd, p 98:40-48 even, p 99: prob. 10, 12, 13, p 100: MR