How do you combine (y-3)/(y-2)-(y+1)/(2y-5)+(-4y+7)/(2y^2-9y+10)?

Jul 16, 2016

$\frac{y - 12}{2 y - 5}$

Explanation:

Since $\left(y - 2\right) \left(2 y - 5\right) = 2 {y}^{2} - 9 y + 10$,

you can obtain an equivalent expression with a single fraction:

$\frac{\left(y - 3\right) \left(2 y - 5\right) - \left(y + 1\right) \left(y - 2\right) + \left(- 4 y + 7\right)}{2 {y}^{2} - 9 y + 10}$

$= \frac{2 {y}^{2} - 5 y - 6 y + 15 - {y}^{2} + 2 y - y + 2 - 4 y + 7}{2 {y}^{2} - 9 y + 10}$

$= \frac{{y}^{2} - 14 y + 24}{2 {y}^{2} - 9 y + 10}$

Then, since

${y}^{2} - 14 y + 24 = \left(y - 2\right) \left(y - 12\right)$,

you can factor:

$\frac{\cancel{y - 2} \left(y - 12\right)}{\cancel{y - 2} \left(2 y - 5\right)}$

so the final expression is:

$\frac{y - 12}{2 y - 5}$