m81291 Solve Equations with Fractions By the end of this section, you will be able to: Determine whether a fraction is a solution of an equation Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality Solve equations using the Multiplication Property of Equality Translate sentences to equations and solve 4a8e9b99-d40c-43c4-a97b-5c3be9cd8750 Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material. Evaluate x+4 when x=−3 If you missed this problem, review . 1 Solve: 2y−3=9. If you missed this problem, review . y=6 Solve: y−3=−9 If you missed this problem, review . -6

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction. Substitute the number for the variable in the equation. Simplify the expressions on both sides of the equation. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. Determine whether each of the following is a solution of x−310=12. ⓐ x=1 ⓑ x=45 ⓒ x=−45 ⓐ Change to fractions with a LCD of 10. Subtract. Since x=1 does not result in a true equation, 1 is not a solution to the equation. ⓑ Subtract. Since x=45 results in a true equation, 45 is a solution to the equation x−310=12. ⓒ Subtract. Since x=−45 does not result in a true equation, −45 is not a solution to the equation. Determine whether each number is a solution of the given equation. x−23=16: ⓐ x=1 ⓑ x=56 ⓒ x=−56 ⓐ no ⓑ yesⓒ no Determine whether each number is a solution of the given equation. y−14=38: ⓐ y=1 ⓑ y=−58 ⓒ y=58 ⓐ no ⓑ noⓒ yes

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.For any numbers a,b, and c, if a=b, then a+c=b+c. Addition Property of Equality if a=b, then a−c=b−c. Subtraction Property of Equality if a=b, then ac=bc,c≠0. Division Property of EqualityIn other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality. Solve: y+916=516. Subtract 916 from each side to undo the addition. Simplify on each side of the equation. Simplify the fraction. Check: Substitute y=−14. Rewrite as fractions with the LCD. Add. Since y=−14 makes y+916=516 a true statement, we know we have found the solution to this equation. Solve: y+1112=512. −12 Solve: y+815=415. −415 We used the Subtraction Property of Equality in . Now we’ll use the Addition Property of Equality. Solve: a−59=−89. Add 59 from each side to undo the subtraction. Simplify on each side of the equation. Simplify the fraction. Check: Substitute a=−13. Change to common denominator. Subtract. Since a=−13 makes the equation true, we know that a=−13 is the solution to the equation. Solve: a−35=−85. −1 Solve: n−37=−97. −67 The next example may not seem to have a fraction, but let’s see what happens when we solve it. Solve: 10q=44. 10q=44 Divide both sides by 10 to undo the multiplication. 10q10=4410 Simplify. q=225 Check: Substitute q=225 into the original equation. 10(225)=?44 Simplify. 102(225)=?44 Multiply. 44=44✓ The solution to the equation was the fraction 225. We leave it as an improper fraction. Solve: 12u=−76. −193 Solve: 8m=92. 232

Consider the equation x4=3. We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal. For any numbers a,b, and c, ifa=b,thenac=bc. If you multiply both sides of an equation by the same quantity, you still have equality. Let’s use the Multiplication Property of Equality to solve the equation x7=−9. Solve: x7=−9. Use the Multiplication Property of Equality to multiply both sides by 7. This will isolate the variable. Multiply. Simplify. The equation is true. Solve: f5=−25. −125 Solve: h9=−27. −243 Solve: p−8=−40. Here, p is divided by −8. We must multiply by −8 to isolate p. Multiply both sides by −8 Multiply. Simplify. Check: Substitute p=320. The equation is true. Solve: c−7=−35. 245 Solve: x−11=−12. 132

Look at the equation −y=15. Does it look as if y is already isolated? But there is a negative sign in front of y, so it is not isolated. There are three different ways to isolate the variable in this type of equation. We will show all three ways in . Solve: −y=15. One way to solve the equation is to rewrite −y as −1y, and then use the Division Property of Equality to isolate y. Rewrite −y as −1y. Divide both sides by −1. Simplify each side. Another way to solve this equation is to multiply both sides of the equation by −1. Multiply both sides by −1. Simplify each side. The third way to solve the equation is to read −y as “the opposite of y.” What number has 15 as its opposite? The opposite of 15 is −15. So y=−15. For all three methods, we isolated y is isolated and solved the equation. Check: Substitute y=−15. Simplify. The equation is true. Solve: −y=48. −48 Solve: −c=−23. 23

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to 1. For example, in the equation: 34x=24 The coefficient of x is 34. To solve for x, we need its coefficient to be 1. Since the product of a number and its reciprocal is 1, our strategy here will be to isolate x by multiplying by the reciprocal of 34. We will do this in . Solve: 34x=24. Multiply both sides by the reciprocal of the coefficient. Simplify. Multiply. Check: Substitute x=32. Rewrite 32 as a fraction. Multiply. The equation is true. Notice that in the equation 34x=24, we could have divided both sides by 34 to get x by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier. Solve: 25n=14. 35 Solve: 56y=15. 18 Solve: −38w=72. The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative. Multiply both sides by the reciprocal of −38. Simplify; reciprocals multiply to one. Multiply. Check: Let w=−192. Multiply. It checks. Solve: −47a=52. −91 Solve: −79w=84. −108

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference. Subtraction Property of Equality:For any real numbers a, b, and c,if a=b, then a−c=b−c. Addition Property of Equality:For any real numbers a, b, and c,if a=b, then a+c=b+c. Division Property of Equality:For any numbers a, b, and c, where c≠0if a=b, then ac=bc Multiplication Property of Equality:For any real numbers a, b, and cif a=b, then ac=bc When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality. In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions. Translate and solve: n divided by 6 is −24. Translate. Multiply both sides by 6. Simplify. Check: Is −144 divided by 6 equal to −24? Translate. Simplify. It checks. Translate and solve: n divided by 7 is equal to −21. n7=−21;n=−147 Translate and solve: n divided by 8 is equal to −56. n8=−56;n=−448 Translate and solve: The quotient of q and −5 is 70. Translate. Multiply both sides by −5. Simplify. Check: Is the quotient of −350 and −5 equal to 70? Translate. Simplify. It checks. Translate and solve: The quotient of q and −8 is 72. q−8=72;q=−576 Translate and solve: The quotient of p and −9 is 81. p−9=81;p=−729 Translate and solve: Two-thirds of f is 18. Translate. Multiply both sides by 32. Simplify. Check: Is two-thirds of 27 equal to 18? Translate. Simplify. It checks. Translate and solve: Two-fifths of f is 16. 25f=16;f=40 Translate and solve: Three-fourths of f is 21. 34f=21;f=28 Translate and solve: The quotient of m and 56 is 34. The quotient of m and 56 is 34. Translate. m56=34 Multiply both sides by 56 to isolate m. 56(m56)=56(34) Simplify. m=5·36·4 Remove common factors and multiply. m=58 Check: Is the quotient of 58 and 56 equal to 34? 5856=?34 Rewrite as division. 58÷56=?34 Multiply the first fraction by the reciprocal of the second. 58·65=?34 Simplify. 34=34✓Our solution checks. Translate and solve. The quotient of n and 23 is 512. n23=512;n=518 Translate and solve The quotient of c and 38 is 49. c38=49;c=16 Translate and solve: The sum of three-eighths and x is three and one-half. Translate. Use the Subtraction Property of Equality to subtract 38 from both sides. Combine like terms on the left side. Convert mixed number to improper fraction. Convert to equivalent fractions with LCD of 8. Subtract. Write as a mixed number. We write the answer as a mixed number because the original problem used a mixed number. Check:Is the sum of three-eighths and 318 equal to three and one-half? 38+318=?312 Add. 348=?312 Simplify. 312=312✓ The solution checks. Translate and solve: The sum of five-eighths and x is one-fourth. 58+x=14;x=−38 Translate and solve: The difference of one-and-three-fourths and x is five-sixths. 134−x=56;x=1112 Solve One Step Equations With Fractions Solve One Step Equations With Fractions by Adding or Subtracting Solve One Step Equations With Fraction by Multiplying

Determine whether a number is a solution to an equation.Substitute the number for the variable in the equation. Simplify the expressions on both sides of the equation. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. Addition, Subtraction, and Division Properties of EqualityFor any numbers a, b, and c, if a=b, then a+c=b+c. Addition Property of Equality if a=b, then a-c=b-c. Subtraction Property of Equality if a=b, then ac=bc, c≠0. Division Property of Equality The Multiplication Property of Equality For any numbers ab and c, a=b, then ac=bc. If you multiply both sides of an equation by the same quantity, you still have equality.

Determine Whether a Fraction is a Solution of an Equation In the following exercises, determine whether each number is a solution of the given equation. x−25=110: ⓐ x=1 ⓑ x=12 ⓒ x=−12 y−13=512: ⓐ y=1 ⓑ y=34 ⓒ y=−34 ⓐ noⓑ yesⓒ no h+34=25: ⓐ h=1 ⓑ h=720 ⓒ h=−720 k+25=56: ⓐ k=1 ⓑ k=1330 ⓒ k=−1330 ⓐ noⓑ yesⓒ no Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality In the following exercises, solve. y+13=43 m+38=78 m=12 f+910=25 h+56=16 h=−23 a−58=−78 c−14=−54 c = −1 x−(−320)=−1120 z−(−512)=−712 z = −1 n−16=34 p−310=58 p=3740 s+(−12)=−89 k+(−13)=−45 k=−715 5j=17 7k=18 k=187 −4w=26 −9v=33 v=−113 Solve Equations with Fractions Using the Multiplication Property of Equality In the following exercises, solve. f4=−20 b3=−9 b = −27 y7=−21 x8=−32 x = −256 p−5=−40 q−4=−40 q = 160 r−12=−6 s−15=−3 s = 45 −x=23 −y=42 y = −42 −h=−512 −k=−1720 k=1720 45n=20 310p=30 p = 100 38q=−48 52m=−40 m = −16 −29a=16 −37b=9 b = −21 −611u=−24 −512v=−15 v = 36 Mixed Practice In the following exercises, solve. 3x=0 8y=0 y = 0 4f=45 7g=79 g=19 p+23=112 q+56=112 q=−34 78m=110 14n=710 n=145 −25=x+34 −23=y+38 y=−2524 1120=-f 815=-d d=−815 Translate Sentences to Equations and Solve In the following exercises, translate to an algebraic equation and solve. n divided by eight is −16. n divided by six is −24. n6=−24;n=−144 m divided by −9 is −7. m divided by −7 is −8. m−7=−8;m=56 The quotient of f and −3 is −18. The quotient of f and −4 is −20. f−4=−20;f=80 The quotient of g and twelve is 8. The quotient of g and nine is 14. g9=14;g=126 Three-fourths of q is 12. Two-fifths of q is 20. 25q=20;q=50 Seven-tenths of p is −63. Four-ninths of p is −28. 49p=−28;p=−63 m divided by 4 equals negative 6. The quotient of h and 2 is 43. h2=43;h=86 Three-fourths of z is 15. The quotient of a and 23 is 34. a23=34;a=12 The sum of five-sixths and x is 12. The sum of three-fourths and x is 18. 34+x=18;x=−58 The difference of y and one-fourth is −18. The difference of y and one-third is −16. y−13=−16;y=16

Shopping Teresa bought a pair of shoes on sale for $48. The sale price was 23 of the regular price. Find the regular price of the shoes by solving the equation 23p=48 Playhouse The table in a child’s playhouse is 35 of an adult-size table. The playhouse table is 18 inches high. Find the height of an adult-size table by solving the equation 35h=18. 30 inches

describes three methods to solve the equation −y=15. Which method do you prefer? Why? Richard thinks the solution to the equation 34x=24 is 16. Explain why Richard is wrong. Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

In the following exercises, name the fraction of each figure that is shaded. 59 In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number. 32=112 In the following exercises, convert the improper fraction to a mixed number. 5815 6311 5811 In the following exercises, convert the mixed number to an improper fraction. 1214 945 495 Find three fractions equivalent to 25. Show your work, using figures or algebra. Find three fractions equivalent to −43. Show your work, using figures or algebra. Answers may vary. In the following exercises, locate the numbers on a number line. 58,43,334,4 14,−14,113,−113,72,−72 In the following exercises, order each pair of numbers, using < or >. −1___−25 −212___−3 >

In the following exercises, simplify. −6384 −90120 −34 −14a14b −8x8y −xy In the following exercises, multiply. 25·813 −13·127 −47 29·(−4532) 6m·411 24m11 −14(−32) 165·158 6 In the following exercises, find the reciprocal. 29 154 415 3 −14 −4 Fill in the chart. Opposite Absolute Value Reciprocal −513 310 94 −12 In the following exercises, divide. 23÷16 4 (−3x5)÷(−2y3) 45÷3 415 8÷83 518÷(−b9) −52b

In the following exercises, perform the indicated operation. 315·178 −5712·4411 −26811 8÷223 823÷1112 8 In the following exercises, translate the English phrase into an algebraic expression. the quotient of 8 and y the quotient of V and the difference of h and 6 Vh−6 In the following exercises, simplify the complex fraction 5845 89−4 −29 n438 −156−112 22 In the following exercises, simplify. 5+165 8·4−523·12 736 8·7+5(8−10)9·3−6·4

In the following exercises, add. 38+28 58 45+15 25+15 35 1532+932 x10+710 x+710 In the following exercises, subtract. 811−611 1112−512 12 45−y5 −3130−730 −1915 32−(32) 1115−515−(−215) 815

In the following exercises, find the least common denominator. 13 and 112 13 and 45 15 815 and 1120 34,16,and510 60 In the following exercises, change to equivalent fractions using the given LCD. 13 and 15, LCD =15 38 and 56, LCD =24 924 and 2024 −916 and 512, LCD =48 13,34 and 45, LCD =60 2060,4560 and 4860 In the following exercises, perform the indicated operations and simplify. 15+23 1112−23 14 −910−34 −1136−1120 −7790 −2225+940 y10−13 3y−1030 25+(−59) 411÷27d 14d11 25+(−3n8)(−29n) (23)2(58)2 256225 (1112+38)÷(56−110) In the following exercises, evaluate. y−45 when ⓐ y=−45 ⓑ y=14 ⓐ −85 ⓑ −1120 6mn2 when m=34 and n=−13

In the following exercises, perform the indicated operation. 413+913 1323 625+735 5811+2411 8111 358+378 91320−41120 5110 2310−1910 21112−1712 113 8611−2911

In the following exercises, determine whether the each number is a solution of the given equation. x−12=16: ⓐ x=1 ⓑ x=23 ⓒ x=−13 ⓐ noⓑ yesⓒ no y+35=59: ⓐ y=12 ⓑ y=5245 ⓒ y=−245 In the following exercises, solve the equation. n+911=411 n=−511 x−16=76 h−(−78)=−25 h=−5140 x5=−10 −z=23 z = −23 In the following exercises, translate and solve. The sum of two-thirds and n is −35. The difference of q and one-tenth is 12. q−110=12;q=35 The quotient of p and −4 is −8. Three-eighths of y is 24. 38y=24;y=64

Convert the improper fraction to a mixed number. 195 Convert the mixed number to an improper fraction. 327 237 Locate the numbers on a number line. 12,123,−234, and 94 In the following exercises, simplify. 520 14 18r27s 13·34 14 35·15 −36u(−49) 16u −5712·4411 −56÷512 −2 711÷(−711) 9a10÷15a8 1225 −625÷4 (−1556)÷(−316) 5 −6611 p2q5 5p2q −415−223 92−429−4 13 2d+9d −313+(−413) −713 −2225+940 25+(−75) −1 −310+(−58) −34÷x3 −94x 23−22(34)2 514+18956 3 Evaluate. x+13 when ⓐ x=23 ⓑ x=−56 In the following exercises, solve the equation. y+35=75 y=45 a−310=−910 f+(−23)=512 f=1312 m−2=−16 −23c=18 c = −27 Translate and solve: The quotient of p and −4 is −8. Solve for p.