$$ \begin{aligned} \frac{y-1}{y-5} &= \frac{y-2}{y-4}&& \text{multiply ALL terms by } \color{blue}{ (y-5)(y-4) }. \\[1 em](y-5)(y-4)\frac{y-1}{y-5} &= (y-5)(y-4)\frac{y-2}{y-4}&& \text{cancel out the denominators} \\[1 em]y^2-5y+4 &= y^2-7y+10&& \text{move all terms to the left hand side } \\[1 em]y^2-5y+4-y^2+7y-10 &= 0&& \text{simplify left side} \\[1 em]y^2-5y+4-y^2+7y-10 &= 0&& \\[1 em]2y-6 &= 0&& \text{ move the constants to the right } \\[1 em]2y &= 6&& \text{ divide both sides by $ 2 $ } \\[1 em]y &= \frac{6}{2}&& \\[1 em]y &= 3&& \\[1 em] \end{aligned} $$

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