Question

solve for y.\\(
\frac{5}{(y + 2)(y - 3)} = 2 + \frac{1}{y - 3}\\)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
\\(y = \square\\)

Explanation:

Step1: Eliminate denominators

Multiply all terms by $(y+2)(y-3)$:
$$5 = 2(y+2)(y-3) + (y+2)$$

Step2: Expand right-hand side

First expand $(y+2)(y-3)$, then distribute and combine:
$$5 = 2(y^2 - 3y + 2y - 6) + y + 2$$
$$5 = 2(y^2 - y - 6) + y + 2$$
$$5 = 2y^2 - 2y - 12 + y + 2$$
$$5 = 2y^2 - y - 10$$

Step3: Rearrange to standard quadratic form

Subtract 5 from both sides:
$$2y^2 - y - 15 = 0$$

Step4: Solve quadratic equation

Use quadratic formula $y=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=2$, $b=-1$, $c=-15$:
$$y=\frac{1\pm\sqrt{(-1)^2-4(2)(-15)}}{2(2)}$$
$$y=\frac{1\pm\sqrt{1+120}}{4}$$
$$y=\frac{1\pm\sqrt{121}}{4}$$
$$y=\frac{1\pm11}{4}$$

Step5: Check for extraneous solutions

Original equation has restrictions $y
eq-2, y
eq3$.

• For $y=\frac{1+11}{4}=3$: Invalid (makes denominator 0)
• For $y=\frac{1-11}{4}=-\frac{10}{4}=-\frac{5}{2}$: Valid

Answer:

$y=-\frac{5}{2}$