$$ \frac{4 x}{x-3} $$ and $$ \frac{x}{x-5} $$

Question

UpStudy AI · Accepted Answer

We want to solve the equation

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

**Step 1. Determine the Domain**

For the fractions to be defined, the denominators must be nonzero. Therefore, we require

$$
x-3 \neq 0 \quad \text{and} \quad x-5 \neq 0
$$

which gives

$$
x \neq 3 \quad \text{and} \quad x \neq 5.
$$

**Step 2. Cross Multiply**

Assuming $$ x $$ avoids the values 3 and 5, we can cross multiply:

$$
4x(x-5) = x(x-3).
$$

**Step 3. Expand and Simplify**

Expand both sides:

$$
4x^2 - 20x = x^2 - 3x.
$$

Bring all terms to one side:

$$
4x^2 - 20x - x^2 + 3x = 0.
$$

Combine like terms:

$$
3x^2 - 17x = 0.
$$

**Step 4. Factor and Solve**

Factor out the common factor $$ x $$:

$$
x(3x - 17) = 0.
$$

So the solutions are:

$$
x = 0 \quad \text{or} \quad 3x - 17 = 0.
$$

Solving the second equation:

$$
3x - 17 = 0 \quad \Longrightarrow \quad 3x = 17 \quad \Longrightarrow \quad x = \frac{17}{3}.
$$

**Step 5. Check the Domain Restrictions**

Recall that $$ x \neq 3 $$ and $$ x \neq 5 $$. Both $$ x = 0 $$ and $$ x = \frac{17}{3} $$ are acceptable.

**Final Answer**

$$
x = 0 \quad \text{or} \quad x = \frac{17}{3}.
$$
The solutions are $$ x = 0 $$ and $$ x = \frac{17}{3} $$.

\( \frac{4 x}{x-3} \) and \( \frac{x}{x-5} \)