CPM Homework Help : PC3 Problem 7-132

Rewrite the following fraction as a sum of two fractions with linear denominators by using partial fraction decomposition.

$\frac{5x-19}{(x-3)(x-5)}$

Step 1:

$\frac{A}{x-3}+\frac{B}{x-5}=\frac{5x-19}{\left(x-3\right)\left(x-5\right)}$

Step 2:

$\frac{A}{x-3}\cdot\frac{x-5}{x-5}+\frac{B}{x-5}\cdot\frac{x-3}{x-3}=\frac{5x-19}{(x-3)(x-5)}$

Step 3:

$Ax-5A+Bx-3B=5x-19$

Step 4:

Solve the system:

$\qquad A+B=5\\ -5A-3B=-19$