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

1. Problem Description

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

2. Solution Steps

First, we want to get all terms with $x$ on one side of the equation and the constant terms on the other side. Let's subtract $\frac{1}{5}x$ from both sides of the equation:

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

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

Next, let's add $9$ to both sides of the equation:

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

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

Now, we need to combine the $x$ terms. To do this, we need a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{5}$. The least common denominator is $15$. So, we rewrite the fractions with the common denominator:

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

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

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

Now, we want to isolate $x$. To do this, we multiply both sides of the equation by $\frac{15}{2}$:

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

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

$$x = 15$$

3. Final Answer

$x = 15$