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13 4C. SOLVING EQUATIONS 4c Solving Equations We can add or subtract the same decimal number from both sides of an equation without affecting the solution, just as we do with fractions. In fact, all of the techniques for solving equations that we have learned work with decimal values. You Try It! EXAMPLE 1. Solve for x: x − 1.35 = −2.6. Solution. To undo subtracting 1.35, add 1.35 to both sides of the equation. x − 1.35 = −2.6 Solve for x: x + 1.25 = 0.6 Original equation. x − 1.35 + 1.35 = −2.6 + 1.35 x = −1.25 Add 1.35 to both sides. Simplify: −2.6 + 1.35 = −1.25. Answer: −0.65 ! You Try It! EXAMPLE 2. Solve for x: −1.2x = −4.08. Solution. To undo multiplying by −1.2, divide both sides of the equation by −1.2. −1.2x = −4.08 −1.2x −4.08 = −1.2 −1.2 x = 3.4 Solve for z: −2.5z = 1.4 Original equation. Divide both sides by −1.2. Simplify: −4.08/(−1.2) = 3.4. Answer: −0.56 ! Combining Operations Sometimes, more than one operation is needed to solve a given equation. You Try It! EXAMPLE 3. Solve for x: −3.8x − 1.7 = −17.28. Solution. To undo subtracting 1.7, add 1.7 to both sides of the equation. −3.8x − 1.7 = −17.28 −3.8x − 1.7 + 1.7 = −17.28 + 1.7 −3.8x = −15.58 Original equation. Add 1.7 to both sides Simplify: −17.28 + 1.7 = −15.58. Next, to undo multiplying by −3.8, divide both sides of the equation by −3.8. Solve for u: −0.02u − 3.2 = −1.75 14 MODULE 4. DECIMALS −15.58 −3.8x = −3.8 −3.8 x = 4.1 Divide both sides by −3.8. Simplify: −15.58/(−3.8) = 4.1. Answer: −72.5 Combining Like Terms Combining like terms with decimal coefficients is done in the same manner as combining like terms with integer coefficients. You Try It! Solve for r: −4.2 + 3.6r − 4.1r = 1.86 EXAMPLE 4. Solve the equation for x: 4.2 − 3.1x + 2x = −7.02. Solution. Combine like terms on the left-hand side of the equation. 4.2 − 3.1x + 2x = −7.02 4.2 − 1.1x = −7.02 4.2 − 1.1x − 4.2 = −7.02 − 4.2 −1.1x = −11.02 −1.1x −11.22 = −1.1 −1.1 x = 10.2 Original equation. Combine like terms: −3.1x + 2x = −1.1x. Subtract 4.2 from both sides. Subtract: −7.02 − 4.2 = −11.22. Divide both sides by −1.1. Divide: −11.22/(−1.1) = 10.2. Thus, the solution of the equation is 10.2. Check. As with all equations, we check our solution by substituting our answer into the original equation. 4.2 − 3.1x + 2x = −7.02 Original equation. 4.2 − 3.1(10.2) + 2(10.2) = −7.02 Substitute 10.2 for x. 4.2 − 31.62 + 20.4 = −7.02 Multiply: 3.1(10.2) = 31.62, 2(10.2) = 20.4. −27.42 + 20.4 = −7.02 Order of Ops: Add, left to right. 4.2 − 31.62 = −27.42. −7.02 = −7.02 Add: −27.42 + 20.4 = −7.02. Answer: −12.12 Because the last line is a true statement, the solution x = 10.2 checks. ! 15 4C. SOLVING EQUATIONS Using the Distributive Property Sometimes the distributive property is needed when solving equations. Distributive Property. Let a, b, and c be any numbers. Then, a(b + c) = ab + ac. You Try It! EXAMPLE 5. Solve the equation for x: −6.3x − 0.4(x − 1.2) = −0.86. Solution. We first distribute the −0.4 times each term in the parentheses, then combine like terms. −6.3x − 0.4(x − 1.2) = −0.86 −6.3x − 0.4x + 0.48 = −0.86 −6.7x + 0.48 = −0.86 Solve for x: −2.5x − 0.1(x − 2.3) = 8.03 Original equation. Distribute. Note that −0.4(−1.2) = 0.48. Combine like terms. Next, subtract 0.48 from both sides, then divide both sides of the resulting equation by −6.7. −6.7x + 0.48 − 0.48 = −0.86 − 0.48 −6.7x = −1.34 −1.34 −6.7x = −6.7 −6.7 x = 0.2 Subtract 0.48 from both sides. Simplify: −0.86 − 0.48 = −1.34. Divide both sides by −6.7. Simplify: −1.34/(−6.7) = 0.2. Answer: −3 ! Rounding Solutions An approximate solution is sometimes adequate. You Try It! EXAMPLE 6. Solve the equation answer to the nearest tenth. 3.1x + 4.6 = 2.5 − 2.2x for x. Round the Solution. We need to isolate the terms containing x on one side of the equation. To isolate the the terms containing x to left hand side of the equation, Solve for x: 4.2x − 1.25 = 3.4 + 0.71x 16 MODULE 4. DECIMALS add 2.2x to both sides. Original equation. 3.1x + 4.6 = 2.5 − 2.2x 3.1x + 4.6 + 2.2x = 2.5 − 2.2x + 2.2x Add 2.2x to both sides. 5.3x + 4.6 = 2.5 Combine terms: 3.1x + 2.2x = 5.3x. To undo adding 4.6, subtract 4.6 from both sides of the equation. 5.3x + 4.6 − 4.6 = 2.5 − 4.6 5.3x = −2.1 Subtract 4.6 from both sides. Simplify: 2.5 − 4.6 = −2.1. To further isolate x, divide both sides of the equation by 5.3. 5.3x −2.1 = 5.3 5.3 x ≈ −0.4 To round the answer to the nearest tenth, we must carry the division out one additional place. 0.39 53)21.00 15 9 5 10 4 77 33 Answer: 1.33 Divide both sides by 5.3. Round solution to nearest tenth. Because the “test digit” is greater than or equal to 5, add 1 to the rounding digit and truncate. Test digit −0. 3 9 Rounding digit Thus, −0.39 ≈ −0.4. Thus, −2.1/5.3 ≈ −0.39. !