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1 Lesson 1.2.5 More Two-Step Equations More Two-Step Equations

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2 Lesson 1.2.5 More Two-Step Equations California Standard: Algebra and Functions 4.1 What it means for you: Key Words: Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. You’ll learn how to deal with fractions in equations, and how to check that your answer is right. fraction isolate check

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3 Lesson 1.2.5 More Two-Step Equations When you have a fraction in an equation, you can think of it as being two different operations that have been merged together. That means it can be solved in the same way as any other two-step equation. x 3 4 × 3 ÷ 4

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4 Fractions Can Be Rewritten as Two Separate Steps Lesson 1.2.5 More Two-Step Equations Fractions can be thought of as a combination of multiplication and division. You might see what is essentially the same expression written in several different ways. For example: All five expressions are the same. x 3 4 3x3x 4 3 x 1 4 1 4 (3 x ) ÷ 4

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5 Deal with a Fraction in an Equation as Two Steps Lesson 1.2.5 More Two-Step Equations Because a fraction can be split into two steps, an equation with a fraction in it can be solved using the two-step method. First split the expression into two separate operations. Then solve as a two-step equation. 3 x = 24 x = 8 x = 6 3 4 3 x ÷ 4 = 6 Multiply both sides by 4 Divide both sides by 3 Here x is first multiplied by 3, 3 x ÷ 4 = 6Write out the equation 3 x ÷ 4 = 6 and the result divided by 4. 3 x ÷ 4 = 6

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6 Example 1 Solution follows… Lesson 1.2.5 More Two-Step Equations Find the value of a when = 6. a 2 3 Solution Split the expression into two operations Solve as a two-step equation a = 6 2 3 2 a ÷ 3 = 6 2 a = 18 a = 9 Write out the equation

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7 Solution follows… Lesson 1.2.5 More Two-Step Equations Example 2 Solution Split the expression into two operations Solve as a two-step equation ( h + 2) ÷ 4 = 3 h + 2 = 12 h = 10 Find the value of h when = 3. h + 2 4 = 3 h + 2 4 The whole expression h + 2 is being divided by 4 — the fraction bar “groups” it. Put it in parentheses here to show that this operation originally took priority. Write out the equation This example has a more complicated numerator:

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8 Guided Practice Solution follows… Lesson 1.2.5 More Two-Step Equations Find the value of the variables in Exercises 1–6. 1. a = 2 3. v = 4 5. 6 = s 2 3 1 2 2 5 2. q = 33 4. r = –8 6. = 6 2c2c 3 3 4 4 1 a = 4 v = 6 s = 15 q = 44 r = –2 c = 9

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9 Check Your Answer by Substituting it Back In Lesson 1.2.5 More Two-Step Equations When you’ve worked out the value of a variable you can check your answer is right by substituting it into the original equation. Once you’ve substituted the value in, evaluate the equation — if the equation is still true then your calculated value is a correct solution.

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10 Lesson 1.2.5 More Two-Step Equations 3 x + 2 = 14 3 x + 2 – 2 = 14 – 2 3 x = 12 3 x ÷ 3 = 12 ÷ 3 x = 4 3 x + 2 = 14, x = 4 3(4) + 2 = 14 12 + 2 = 12 14 = 14 First solve the equation to find the value of x. Then evaluate the equation using your calculated value. As both sides are the same, the value of x is correct. Now substitute the calculated value back into the equation.

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11 Example 3 Solution follows… Lesson 1.2.5 More Two-Step Equations Check that c = 8 is a solution of the equation 10 c + 15 = 95. Solution Substitute 8 into the equation Simplify 10 c + 15 = 95 95 = 95 Write out the equation 10(8) + 15 = 95 80 + 15 = 95 The equation is still true, so c = 8 is a solution of the equation 10 c + 15 = 95.

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12 Guided Practice Solution follows… Lesson 1.2.5 More Two-Step Equations Solve the equations below then check that your answers are correct. 7. 12 m + 8 = 56 9. 56 = 18 + 19 v 11. 3 – 6 x = 9 8. 22 + 3 h = 34 10. 16 – 4 g = –28 12. 5 y – 12 = 28 m = 4 v = 2 x = –1 h = 4 g = 11 y = 8

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13 Independent Practice Solution follows… Lesson 1.2.5 More Two-Step Equations Find the value of the variables in Exercises 1–6. 1. d = 24 3. – b = 14 5. 22 = n 2. k = 8 4. 27 = w 6. = 4 2 3 3 4 2 5 4 5 3 2 5t5t 10 d = 32 b = –21 n = 55 k = 10 w = 18 t = 8

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14 Independent Practice Solution follows… Lesson 1.2.5 More Two-Step Equations Solve the equations in Exercises 7–10 and check your solutions. 7. 2 x + 4 = 16 9. 6 = v ÷ 4 + 2 8. 3 r – 6 = –12 10. c = 15 3 4 11. For each of the equations, say whether a) y = 3, or b) y = –3, is a correct solution. Equation 1: 10 – 2 y = 16 Equation 2: – y = –2 2 3 x = 6 v = 16 r = –2 c = 20 b) is a correct solution, a) is not.a) is a correct solution, b) is not.

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15 Independent Practice Solution follows… Lesson 1.2.5 More Two-Step Equations For each equation in Exercises 12–14, say whether the solution given is a correct one. 12. x ÷ 2 + 4 = 9, x = 10. 13. 3 x – 9 = 12, x = 4. 14. 8 = 5 x – 7, x = 3. Yes. No ( x = 7 would be correct). Yes.

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16 Round Up Lesson 1.2.5 More Two-Step Equations You can think of a fraction as a combination of two operations. So a fraction in an equation can be treated as two steps. And don’t forget — when you’ve found a solution, you should always substitute it back into the equation to check that it’s right.

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