$$ \begin{aligned} \frac{5}{y+3}+\frac{2}{y-2} &= \frac{7}{y+5}&& \text{multiply ALL terms by } \color{blue}{ (y+3)(y-2)(y+5) }. \\[1 em](y+3)(y-2)(y+5)\cdot\frac{5}{y+3}+(y+3)(y-2)(y+5)\cdot\frac{2}{y-2} &= (y+3)(y-2)(y+5)\cdot\frac{7}{y+5}&& \text{cancel out the denominators} \\[1 em]5y^2+15y-50+2y^2+16y+30 &= 7y^2+7y-42&& \text{simplify left side} \\[1 em]7y^2+31y-20 &= 7y^2+7y-42&& \text{move all terms to the left hand side } \\[1 em]7y^2+31y-20-7y^2-7y+42 &= 0&& \text{simplify left side} \\[1 em]7y^2+31y-20-7y^2-7y+42 &= 0&& \\[1 em]24y+22 &= 0&& \text{ move the constants to the right } \\[1 em]24y &= -22&& \text{ divide both sides by $ 24 $ } \\[1 em]y &= -\frac{22}{24}&& \\[1 em]y &= -\frac{11}{12}&& \\[1 em] \end{aligned} $$

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