Download Lesson 8: Equations with Fractions - the Algebra Class E

Unit 1: Solving Equations
Lesson 8: Equations with Fractions
Fractions are not easy numbers to deal with, so if your equation contains fractions, your first
step will be to eliminate the fractions.
Steps to eliminate the fraction:
1. If you have only 1 fraction (or 2 or more fractions with the same denominator) , multiply
ALL terms by the denominator. This will eliminate the fraction, and keep your equation
balanced.
2. If you have 2 or more fractions (with different denominators), multiply ALL terms by the
least common denominator (lcd).
Example 1
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Unit 1: Solving Equations
Example 2
Example 3
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Unit 1: Solving Equations
Lesson 8: Solving Equations with Fractions
1.
3.
2.
⅓(x +4) = -6
4. 5x + ½(x – 3) = 15
5. 2x – ⅔(x – 2) = 9
6. ¾ x – 9 +¼(x -2) = 23
7. -20 = ⅜x – ⅝(x – 2) +4
8. -2x + ⅓( x – 3) = -10
9. ¼(2x – 4) = 18
10. ¾ (3y -4) + ¼(2y - 2) = 28
11. 3y - ½(2y -6) -3y = -21
12.
13. 2/3(y – 5) – ¼(4 – y) = 3
14. j – 5/9j = 4
For problems, 15 – 18, decide whether the given solution is correct. If not, find the correct solution.
15. 1/6x – 3(x – 5) = -36
16. 1/5 – 2/5(y + 10) +1/10 = -1.7
x = 25.5
y = -5
17. 1/3(p – 8) + ½(p +2) = 12.5
P = 17
18. 3/5 – 2/3(y – 7) = -1.4
y = -4
Directions: Solve each equation. (2 points each)
1.
2. .
3. -4x – ½(x + 12) = 23.25
4. ¾ x + 18 -¼(x - 9) = 21.75
**5. ½(x-4) + ¾(x+ 5) = 13
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2/3(x - 8) = 6
Unit 1: Solving Equations
Lesson 8: Solving Equations with Fractions
Answer Key
1.
2.
.
x +4 = 20
2x -2 = -36
x +4 -4 = 20 -4
2x -2 +2 = -36 +2
x = 16
2x = -34
2
2
x = -17
3. ⅓(x +4) = -6
3[⅓(x +4)] = -6(3)
4. 5x + ½(x – 3) = 15
2[5x + ½(x – 3)] = 15(2)
1(x+4) = -18
10x +1(x -3) = 30
x +4 = -18
10x +x -3 = 30
x +4 -4 = -18 -4
11x -3 = 30
x = -22
11x -3 +3 = 30 +3
11x = 33
11
11
x=3
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Unit 1: Solving Equations
5. 2x – ⅔(x – 2) = 9
6. ¾ x – 9 +¼(x -2) = 23
3[2x – ⅔(x – 2)] = 9(3)
4[¾ x – 9 +¼(x -2)] = 23(4)
6x -2(x -2) = 27
3x – 36 +1(x -2) = 92
6x – 2x +4 = 27
3x – 36 +x -2 = 92
4x +4 = 27
3x +x – 36 – 2 = 92
4x +4 -4 = 27 -4
4x -38 = 92
4x = 23
4
4
4x – 38 +38 = 92 +38
x = 23/4
4x = 130
4
4
x = 32.5
7. -20 = ⅜x – ⅝(x – 2) +4
8. -2x + ⅓( x – 3) = -10
⅜x – ⅝(x – 2) +4 = -20
3[-2x + ⅓( x– 3)] = -10(3)
8[⅜x – ⅝(x – 2) +4] = -20(8)
-6x +1(x -3) = -30
3x – 5(x-2) + 32 = -160
-6x + x -3 = -30
3x – 5x +10 +32 = -160
-5x -3 +3 = -30 +3
-2x + 42 = -160
-2x +42 -42 = -160 – 42
-5x = -27
-5
-5
-2x = -202
-2
-2
x = 27/5
x = 101
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Unit 1: Solving Equations
9. ¼(2x – 4) = 18
10. ¾ (3y -4) + ¼(2y - 2) = 28
4[¼(2x – 4)] = 18(4)
4[¾ (3y -4) + ¼(2y - 2)] = 28(4)
1(2x -4) = 72
3(3y -4) +1(2y -2) = 112
2x -4 = 72
9y -12 +2y -2 = 112
2x -4 +4 = 72 +4
9y +2y – 12 -2 = 112
2x = 76
2
2
11y – 14 = 112
11y – 14 +14 = 112 +14
x = 38
11y = 126
11
11
y = 126/11
11. 3y - ½(2y -6) -3y = -21
12.
2[3y - ½(2y -6) -3y] = -21(2)
6y – 1(2y -6) – 6y = -42
x +4 + 1(2x -4) = 24
6y -2y +6 -6y = -42
x +4 +2x -4 = 24
6y-2y -6y +6 = -42
x +2x +4 -4 = 24
-2y +6 = -42
3x +0 = 24
-2y +6 -6 = -42 -6
-2y = -48
-2
-2
3x = 24
3
3
x=8
y = 24
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Unit 1: Solving Equations
13. 2/3(y – 5) – ¼(4 – y) = 3
12[2/3(y-5) – ¼(4-y) = 3(12)
14. j – 5/9j = 4
9[j – 5/9j] = 4(9)
8(y-5) – 3(4-y) = 36
9j – 5j = 36
8y – 40 – 12 +3y = 36
4j = 36
8y + 3y -40 – 12 = 36
4j/4 = 36/4
11y -52 = 36
j=9
11y – 52 + 52 = 36 + 52
11y = 88
11y /11 = 88/11
y=8
For problems, 15 – 18, decide whether the given solution is correct. If not, find the correct
solution.
15. 1/6x – 3(x – 5) = -36
16. 1/5 – 2/5(y + 10) +1/10 = -1.7
x = 25.5
y = -5
NO: x = 25.5 is not correct.
YES, y = -5 is correct
6[1/6x – 3(x-5) = -36(6)
1/5 – 2/5(-5+10) +1/10 = -1.7
x – 18(x-5) = -216
.2 - .4(5)+.1 = -1.7
x – 18x + 90 = -216
.2 – 2 +.1 = -1.7
-17x + 90 -90 = -216 – 90
-1.7 = -1.7
-17x = -306
-17x/-17 = -306/-17
x = 18
17. 1/3(p – 8) + ½(p +2) = 12.5
P = 17
YES; p = 17 is correct
1/3(17 – 8) + ½(17 + 2) = 12.5
1/3(9) + ½(19) = 12.5
3 + 9.5 = 12.5
12.5 = 12.5
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18. 3/5 – 2/3(y – 7) = -1.4
y = -4
NO, y = -4 is not correct.
15[3/5 – 2/3(y-7) = -1.4(15)
9 – 10(y-7) = -21
9 – 10y + 70 = -21
-10y +79 = -21
-10y + 79 – 79 = -21 -79
-10y = -100
-10y /10 = -100/-10
y = 10
Unit 1: Solving Equations
Directions: Solve each equation. Show every step. (2 points each)
1.
Multiply by 3 on both sides
2.
3x – 2 = -54
Simplify: -18 · 3 = -54
3x – 2 + 2 = -54 +2
Add 2 to both sides
3x = -52
Simplify: -54+2 = -52
3x /3 = -52/3
Divide by 3 on both sides
x = -52/3
Simplify (Leave you answer as a fraction)
2/3(x - 8) = 6
3[2/3(x-8)] = 6(3)
Multiply by 3 on both sides to get rid of fraction
2(x-8) = 18
Simplify
2x-16 = 18
Distribute 2 throughout the parenthesis
2x-16+16 = 18+16
Add 16 to both sides
2x = 34
Simplify: 18+16 = 34
2x/2 = 34/2
Divide by 2 on both sides
x = 17
Simplify: 34/2 = 17
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Unit 1: Solving Equations
3. -4x – ½(x + 12) = 23.25
2[-4x – ½(x+12)] = 23.25(2)
Multiply by 2 to get rid of the fraction
-8x – (x+12) = 46.5
Simplify
-8x – x – 12 = 46.5
Distribute -1 throughout the parenthesis
-9x – 12 = 46.5
Combine like terms: -8x – x = 9x
-9x – 12 +12 = 46.5+12
Add 12 to both sides
--9x = 58.5
Simplify: 46.5+12 = 58.5
-9x/-9 = 58.5/-9
Divide by -9 on both sides
x = -6.5
Simpify: 58.5/-9 = -6.5
4. ¾ x + 18 -¼(x - 9) = 21.75
4[3/4x+18-1/4(x-9)] = 21.75(4)
Multiply by 4 to get rid of the fractions
3x + 72 – (x-9) = 87
Simplify
3x + 72 – x + 9 = 87
Distribute -1 throughout the parenthesis
3x –x +72+9= 87
Rewrite like terms together
2x +81=87
Combine like terms; 3x-x = 2x & 72+9=81
2x +81 -81 = 87-81
Subtract 81 from both sides
2x = 6
Simplify: 87-81=6
2x2 = 6/2
Divide by 2 on both sides
x=3
Simplify: 6/2=3
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Unit 1: Solving Equations
**5. ½(x-4) + ¾(x+ 5) = 13
4[1/2(x-4) + ¾(x+5)] = 13(4)
Multiply all terms by 4 to get rid of both
fractions
2(x-4) + 3(x+5) = 52
Simplify
2x-8 + 3x+15 = 52
Distribute the 2 and 3 throughout the
parenthesis
2x + 3x -8 +15 = 52
Rewrite like terms together
5x + 7 = 52
Combine like terms
5x+7-7 = 52-7
Subtract 7 from both sides
5x = 45
Simplify: 52-7 = 45
5x/5 = 45/5
Divide by 5 on both sides
x=9
Simplify: 45/5 = 9
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