Question

solve the following rational equation and simplify your answer.
\\(\frac{y}{y - 2} - \frac{8}{y + 5} = \frac{y^2}{y^2 + 3y - 10}\\)

Explanation:

Step1: Factor the right-hand denominator

First, factor $y^2 + 3y - 10$:
$y^2 + 3y - 10 = (y-2)(y+5)$
The equation becomes:
$\frac{y}{y-2} - \frac{8}{y+5} = \frac{y^2}{(y-2)(y+5)}$

Step2: Eliminate denominators

Multiply all terms by the common denominator $(y-2)(y+5)$ (note $y
eq 2, -5$ to avoid division by zero):
$y(y+5) - 8(y-2) = y^2$

Step3: Expand each term

Expand the left-hand side:
$y^2 + 5y - 8y + 16 = y^2$

Step4: Simplify and solve for y

Combine like terms and isolate $y$:
$y^2 - 3y + 16 = y^2$
Subtract $y^2$ from both sides:
$-3y + 16 = 0$
$-3y = -16$
$y = \frac{16}{3}$

Step5: Verify the solution

Check that $y=\frac{16}{3}$ does not make any original denominator zero (it does not, since $\frac{16}{3}
eq 2, -5$), so it is valid.

Answer:

$y = \frac{16}{3}$