This document provides an overview of solving multi-step equations with fractions and decimals. It discusses identifying coefficients, multiplying terms by the reciprocal of fractions to clear them, and two methods for working with decimal coefficients - either keeping them as decimals or clearing them by multiplying all terms by a power of 10. Sample problems are provided and worked through as examples.

1. Chapter 7 Section 3: Multi-Step Equations with Fractions and Decimals January 7 th , 2009

2. Warm Up: -9 = 3(y + 4) 5(t – 8) = 10 x + 7 – 3x = 7 y = -7 t = 10 x = 0

3. Coefficients of Variables A coefficient is the number, fraction, or decimal INFRONT OF a variable . When the coefficient is a fraction , use the RECIPROCAL of that fraction to solve the equation. Remember, always MULTIPLY by the reciprocal. Solve: (4/5)x = 12 This is a Coefficient This is a Constant

4. Multi-Step Equations & Fractions/Coefficients Still follow the rules of any Multi-Step Equation, UNDO the Equation, SADMEP! Get all of the constants on one side, leaving the variables and coefficients on the other side. (coefficient)(variable) = constant Then get rid of the coefficient by undoing it. For example: multiplying by the reciprocal.

5. Like This Solve (2/3)n – 6 = 22 (2/3)n – 6 + 6 = 22 + 6 (Add. Prop. of Eq.) (3/2) (2/3)n = 28 (3/2) (Multiply by the Reciprocal) (6/6)n = (28 • 3)/2 (Simplify) 1n = 84/2 (Simplify) n = 42 (Solved)

6. Try These: Undo (SADMEP) (-7/10)k + 14 = - 21 (2/3)(m – 6) = 3 50 = k (21/2) or (10 ½) = m

7. Real World Problem A student has two test scores of 93 and 80. Solve the equation (93+80+t)/3 = 90 to find what the student would have to score on a third test to average 90.

8. Step By Step The student would have to score 97 on a third test to average 90. ( 93 + 80 + t ) = 90 Equation 3 (3) ( 93 + 80 + t ) = 90 (3) Multi. Prop. of Eq. 3 93 + 80 + t = 270 Simplify 173 + t = 270 Simplify 173 – 173 + t = 270 – 173 Sub. Prop. of Eq. t = 97 Simplify

9. Try These - 12 + x = 13 2 4 (a + 6) = 2 7 -38 = x -5/2, or (–2 ½) = a

10. Use the LEAST COMMON MULTIPLE (LCM) of the Denominator to clear the equation of fractions. 2 x + 2 = 3 Find a LCM for 5 and 4 . 5 4 20 (2/5 x + 2) = 20 (3/4) Multi. Prop. of Equality 40/5 x + 40 = 60/4 Simplify 8x + 40 = 15 Simplify Again 8x + 40 – 40 = 15 – 40 Sub. Prop. of Equality 8x = - 25 Div. Prop of Equality x = -25/8 or (–3 1/8) Solution Making Fractions Easier

11. Try These -(5/8)y + y = 1/8 (1/3)b – 1 = 5/6 1/3 = y 11/2, or 5 ½ = b

12. Working with Decimals If you have decimals as coefficients, you can work with them two ways. Method 1 : Just working with the decimal as a decimal. Method 2 : Clear out the decimal by multiplying everything by powers 10.

13. Method 1: Dismal Decimals Method 1: Just work with them. 0.035m + 9.95 = 12.75 - 9.95 -9.95 Sub. Prop. of Eq. 0.035m = 2.8 0.035 0.035 Div. Prop. of Eq. m = 80 Just have to do long division with decimals. Not too bad…BUT, Method 2, might be easier.

14. Method 2: Clear the Decimals Use the decimal with the greatest number of decimal places to decide what power of 10 to use. So if the greatest decimal is 0.0001, multiply everything by 10,000. 0.035m + 9.95 = 12.75 1,000 (0.035m + 9.95) = 12.75 (1,000) 35m + 9,950 – 9,950 = 12,750 – 9,950 35m /35 = 2,800 /35 m = 80

15. Try This One With Either Method -0.8k – 3.1 = -8.3 Answer, k = 6.5

16. Assignment #54 Page 348: 11-35 all.