Let's solve each of these linear equations step by step. I'll go through each one carefully, especially since they involve fractions.
---
1.
$$
\frac{3x}{4} = \frac{1}{8}
$$
Step 1: Multiply both sides by 4 to eliminate the denominator on the left:
$$
3x = \frac{4}{8} = \frac{1}{2}
$$
Step 2: Divide both sides by 3:
$$
x = \frac{1}{2} \div 3 = \frac{1}{6}
$$
✔ Answer: $ \boxed{\frac{1}{6}} $
---
2.
$$
\frac{5x}{7} - \frac{x}{14} = \frac{9}{7}
$$
Step 1: Find a common denominator for the left side (LCM of 7 and 14 is 14):
$$
\frac{10x}{14} - \frac{x}{14} = \frac{9}{7}
\Rightarrow \frac{9x}{14} = \frac{9}{7}
$$
Step 2: Multiply both sides by 14:
$$
9x = 18
\Rightarrow x = 2
$$
✔ Answer: $ \boxed{2} $
---
3.
$$
\frac{2}{3}x - \frac{1}{6} = \frac{5}{6}x - \frac{2}{3}
$$
Step 1: Eliminate fractions by multiplying every term by the LCD of denominators: LCM of 3 and 6 is 6.
Multiply all terms by 6:
$$
6 \cdot \left( \frac{2}{3}x \right) - 6 \cdot \left( \frac{1}{6} \right) = 6 \cdot \left( \frac{5}{6}x \right) - 6 \cdot \left( \frac{2}{3} \right)
\Rightarrow 4x - 1 = 5x - 4
$$
Step 2: Solve:
$$
4x - 1 = 5x - 4 \\
-1 + 4 = 5x - 4x \\
3 = x
$$
✔ Answer: $ \boxed{3} $
---
4.
$$
\frac{5}{12}x - \frac{3}{4} = \frac{7}{6} - \frac{2}{9}x
$$
Step 1: Find LCD of denominators: 12, 4, 6, 9 → LCM is 36.
Multiply every term by 36:
$$
36 \cdot \left( \frac{5}{12}x \right) - 36 \cdot \left( \frac{3}{4} \right) = 36 \cdot \left( \frac{7}{6} \right) - 36 \cdot \left( \frac{2}{9}x \right)
\Rightarrow 15x - 27 = 42 - 8x
$$
Step 2: Solve:
$$
15x + 8x = 42 + 27 \\
23x = 69 \\
x = 3
$$
✔ Answer: $ \boxed{3} $
---
5.
$$
\frac{4}{7} + \frac{3}{7}x = \frac{20}{7} - \frac{5}{7}x
$$
Step 1: Multiply every term by 7 to eliminate denominators:
$$
4 + 3x = 20 - 5x
$$
Step 2: Solve:
$$
3x + 5x = 20 - 4 \\
8x = 16 \\
x = 2
$$
✔ Answer: $ \boxed{2} $
---
6.
$$
\frac{4}{5}x - \frac{3}{7} = \frac{4}{7}x + \frac{5}{7}
$$
Step 1: Multiply every term by LCM of 5 and 7 = 35:
$$
35 \cdot \left( \frac{4}{5}x \right) - 35 \cdot \left( \frac{3}{7} \right) = 35 \cdot \left( \frac{4}{7}x \right) + 35 \cdot \left( \frac{5}{7} \right)
\Rightarrow 28x - 15 = 20x + 25
$$
Step 2: Solve:
$$
28x - 20x = 25 + 15 \\
8x = 40 \\
x = 5
$$
✔ Answer: $ \boxed{5} $
---
7.
$$
\frac{3x - 1}{2} - \frac{x - 2}{3} = 6
$$
Step 1: Multiply every term by LCD of 2 and 3 = 6:
$$
6 \cdot \left( \frac{3x - 1}{2} \right) - 6 \cdot \left( \frac{x - 2}{3} \right) = 6 \cdot 6 \\
\Rightarrow 3(3x - 1) - 2(x - 2) = 36
$$
Step 2: Expand:
$$
9x - 3 - 2x + 4 = 36 \\
7x + 1 = 36 \\
7x = 35 \\
x = 5
$$
✔ Answer: $ \boxed{5} $
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8.
How many solutions are there?
$$
4x - 1 = 4x - 1
$$
Step 1: Subtract $4x$ from both sides:
$$
-1 = -1
$$
This is always true, regardless of $x$. So it's an identity.
✔ Answer: $ \boxed{\text{C) many solutions}} $
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✔ Final Answers Summary:
1. $ \boxed{\frac{1}{6}} $
2. $ \boxed{2} $
3. $ \boxed{3} $
4. $ \boxed{3} $
5. $ \boxed{2} $
6. $ \boxed{5} $
7. $ \boxed{5} $
8. $ \boxed{\text{C) many solutions}} $
Let me know if you'd like explanations in a different format or want to check more problems!
Parent Tip: Review the logic above to help your child master the concept of solving linear equations with fractions worksheet.
