Let’s solve each of these multi-step equations involving fractions step by step. The goal is to isolate the variable on one side of the equation.
---
1) 3(a - 2/3) = (3/4)a + 2 1/4
First, convert mixed number:
2 1/4 = 9/4
Distribute left side:
3a - 2 = (3/4)a + 9/4
Subtract (3/4)a from both sides:
3a - (3/4)a - 2 = 9/4
→ (12/4 - 3/4)a - 2 = 9/4
→ (9/4)a - 2 = 9/4
Add 2 to both sides:
(9/4)a = 9/4 + 2 = 9/4 + 8/4 = 17/4
Multiply both sides by 4/9:
a = (17/4) * (4/9) = 17/9
✔ Answer: a = 17/9
---
2) (7/2) - (3/5) = (-2/3)z + (1/6)
Left side: find common denominator for 7/2 and 3/5 → LCD = 10
7/2 = 35/10, 3/5 = 6/10 → 35/10 - 6/10 = 29/10
Equation becomes:
29/10 = (-2/3)z + 1/6
Subtract 1/6 from both sides:
29/10 - 1/6 = (-2/3)z
LCD of 10 and 6 is 30:
29/10 = 87/30, 1/6 = 5/30 → 87/30 - 5/30 = 82/30 = 41/15
So:
41/15 = (-2/3)z
Multiply both sides by reciprocal of -2/3 → multiply by -3/2:
z = (41/15) * (-3/2) = -123/30 = -41/10
✔ Answer: z = -41/10
---
3) (7/4)x - 3 = 2 + (9/4)x
Subtract (9/4)x from both sides:
(7/4)x - (9/4)x - 3 = 2
→ (-2/4)x - 3 = 2 → (-1/2)x - 3 = 2
Add 3 to both sides:
(-1/2)x = 5
Multiply both sides by -2:
x = -10
✔ Answer: x = -10
---
4) (3c + 8)/3 = 1/2 + c/4
Multiply both sides by LCD of denominators (3, 2, 4) → LCD = 12:
12 * [(3c + 8)/3] = 12 * [1/2 + c/4]
→ 4*(3c + 8) = 6 + 3c
→ 12c + 32 = 6 + 3c
Subtract 3c from both sides:
9c + 32 = 6
Subtract 32:
9c = -26
Divide by 9:
c = -26/9
✔ Answer: c = -26/9
---
5) 1/3 - (2/9)m = 15 + m
Get all terms with m on one side. Subtract m from both sides:
1/3 - (2/9)m - m = 15
→ 1/3 - (2/9 + 9/9)m = 15
→ 1/3 - (11/9)m = 15
Subtract 1/3 from both sides:
-(11/9)m = 15 - 1/3 = 45/3 - 1/3 = 44/3
Multiply both sides by -9/11:
m = (44/3) * (-9/11) = (44 * -9) / (3 * 11) = (4 * -9) / 3 = -36/3 = -12
✔ Answer: m = -12
---
6) (1/2)(q + 1) = 4/3 - q
Distribute left:
(1/2)q + 1/2 = 4/3 - q
Add q to both sides:
(1/2)q + q + 1/2 = 4/3
→ (3/2)q + 1/2 = 4/3
Subtract 1/2:
(3/2)q = 4/3 - 1/2 = 8/6 - 3/6 = 5/6
Multiply both sides by 2/3:
q = (5/6) * (2/3) = 10/18 = 5/9
✔ Answer: q = 5/9
---
7) (1/6)r + 2 = 4 1/6 r + 8/3
Convert 4 1/6 to improper fraction: 25/6
Equation:
(1/6)r + 2 = (25/6)r + 8/3
Subtract (1/6)r from both sides:
2 = (24/6)r + 8/3 → 2 = 4r + 8/3
Subtract 8/3:
2 - 8/3 = 4r → 6/3 - 8/3 = -2/3 = 4r
Divide by 4:
r = (-2/3) / 4 = -2/12 = -1/6
✔ Answer: r = -1/6
---
8) (2/3) - (3/2)y + (1/3)y + 4 = 0
Combine like terms. First, combine y terms:
- (3/2)y + (1/3)y = (-9/6 + 2/6)y = (-7/6)y
Constants: 2/3 + 4 = 2/3 + 12/3 = 14/3
Equation:
(-7/6)y + 14/3 = 0
Subtract 14/3:
(-7/6)y = -14/3
Multiply both sides by -6/7:
y = (-14/3) * (-6/7) = (14 * 6) / (3 * 7) = (2 * 6) / 3 = 12/3 = 4
✔ Answer: y = 4
---
##
✔ Final Answers:
1)
a = 17/9
2)
z = -41/10
3)
x = -10
4)
c = -26/9
5)
m = -12
6)
q = 5/9
7)
r = -1/6
8)
y = 4
Let me know if you’d like to see any step explained in more detail!
Parent Tip: Review the logic above to help your child master the concept of solving multi step equations with fractions worksheet.
