We need to solve the linear equation $\frac{1}{3}x - 9 = \frac{1}{5}x - 7$ for $x$.

1. Problem Description

We need to solve the linear equation $\frac{1}{3}x - 9 = \frac{1}{5}x - 7$ for $x$.

2. Solution Steps

First, add 9 to both sides of the equation to isolate the x terms on one side and the constants on the other.

$\frac{1}{3}x - 9 + 9 = \frac{1}{5}x - 7 + 9$

$\frac{1}{3}x = \frac{1}{5}x + 2$

Next, subtract $\frac{1}{5}x$ from both sides of the equation:

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

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

Find a common denominator for the fractions, which is

1

5. $\frac{5}{15}x - \frac{3}{15}x = 2$

$\frac{5-3}{15}x = 2$

$\frac{2}{15}x = 2$

Now, multiply both sides of the equation by $\frac{15}{2}$ to isolate x.

$\frac{15}{2} \cdot \frac{2}{15}x = 2 \cdot \frac{15}{2}$

$x = 15$

3. Final Answer

$x = 15$