Solve the following equations for $x$: Problem 7: $\frac{4}{3x} - \frac{3}{4x} = 7$ Problem 9: $\frac{5x}{4} - \frac{x-1}{2} = \frac{x-1}{3}$

1. Problem Description

Solve the following equations for $x$:

Problem 7: $\frac{4}{3x} - \frac{3}{4x} = 7$

Problem 9: $\frac{5x}{4} - \frac{x-1}{2} = \frac{x-1}{3}$

2. Solution Steps

Problem 7:

$\frac{4}{3x} - \frac{3}{4x} = 7$

Find a common denominator, which is $12x$.

$\frac{4 \cdot 4}{3x \cdot 4} - \frac{3 \cdot 3}{4x \cdot 3} = 7$

$\frac{16}{12x} - \frac{9}{12x} = 7$

$\frac{16-9}{12x} = 7$

$\frac{7}{12x} = 7$

Multiply both sides by $12x$:

$7 = 7 \cdot 12x$

Divide both sides by 7:

$1 = 12x$

Divide by 12:

$x = \frac{1}{12}$

Problem 9:

$\frac{5x}{4} - \frac{x-1}{2} = \frac{x-1}{3}$

Find a common denominator, which is

1

2. $\frac{5x \cdot 3}{4 \cdot 3} - \frac{(x-1) \cdot 6}{2 \cdot 6} = \frac{(x-1) \cdot 4}{3 \cdot 4}$

$\frac{15x}{12} - \frac{6(x-1)}{12} = \frac{4(x-1)}{12}$

Multiply both sides by 12:

$15x - 6(x-1) = 4(x-1)$

$15x - 6x + 6 = 4x - 4$

$9x + 6 = 4x - 4$

$9x - 4x = -4 - 6$

$5x = -10$

$x = \frac{-10}{5}$

$x = -2$

3. Final Answer

Problem 7: $x = \frac{1}{12}$

Problem 9: $x = -2$