CPM Homework Help : PC3 Problem 5-119

Samantha and Jeanne each create a rational expression with a linear factor in the denominator and give the sum of their expressions to Sarah and Karen, challenging them to figure out the original fractions. If the sum is $\frac { - 9 x - 41 } { ( 2 x + 3 ) ( x - 4 ) }$ what were Samantha and Jeanne’s original fractions?

Step 1:

$\frac{A}{2x+3}+\frac{B}{x-4}=\frac{-9x-41}{(2x+3)(x-4)}$

Step 2:

$\frac{A(x-4)}{(2x+3)(x-4)}+\frac{B(2x+3)}{(x-4)(2x+3)}=\frac{-9x-41}{(2x+3)(x-4)}$

Step 3:

Rewrite as a system and solve: $A\left(x-4\right)+B\left(2x+3\right)=-9-41$