Solve the Rational Equation: 7/(x-5) = 15/(3-x)

To solve this problem, we'll follow these steps:

• Step 1: Transform the equation to ensure the denominators are handled correctly.
• Step 2: Apply cross-multiplication to eliminate the fractions.
• Step 3: Simplify the resulting equation and solve for $x$.
• Step 4: Verify that the solution does not make either denominator zero.

Now, let's work through each step:

Step 1: Recognize that $3-x = -(x-3)$, so we can rewrite the equation as:

$\frac{7}{x-5}=\frac{15}{-(x-3)}$, which simplifies to $\frac{7}{x-5} = -\frac{15}{x-3}$.

Step 2: Apply cross-multiplication:

Multiply both sides to clear the fractions:

$7 \cdot (x-3) = -15 \cdot (x-5)$.

Step 3: Distribute and solve for $x$:

Expanding both sides, we get: $7x - 21 = -15x + 75$.

Bring all terms involving $x$ to one side:

$7x + 15x = 75 + 21$.

This simplifies to:

$22x = 96$.

Now, solve for $x$:

$x = \frac{96}{22}$.

Simplify the fraction:

$x = \frac{48}{11}$.

Convert to a decimal, if preferred:

$x \approx 4.36$.

Step 4: Verify that the solution does not make either denominator zero:

With $x = 4.36$, neither $x - 5$ nor $3 - x$ is zero, so the solution is valid.

Therefore, the solution to the equation is $\boxed{4.36}$.