Solve x+2/(x+5)(x+3)=A/(x+5)+B/(x+3)

$$\frac { x + 2 } { ( x + 5 ) ( x + 3 ) } = \frac { A } { ( x + 5 ) } + \frac { B } { ( x + 3 ) }$$

Solve for A

$$A=-\frac{Bx-x+5B-2}{x+3}$$

$$x\neq -5\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of \left(x+5\right)\left(x+3\right),x+5,x+3.

$$x+2=\left(x+3\right)A+\left(x+5\right)B$$

Use the distributive property to multiply x+3 by A.

$$x+2=xA+3A+\left(x+5\right)B$$

Use the distributive property to multiply x+5 by B.

$$x+2=xA+3A+xB+5B$$

Swap sides so that all variable terms are on the left hand side.

$$xA+3A+xB+5B=x+2$$

Subtract xB from both sides.

$$xA+3A+5B=x+2-xB$$

Subtract 5B from both sides.

$$xA+3A=x+2-xB-5B$$

Combine all terms containing A.

$$\left(x+3\right)A=x+2-xB-5B$$

The equation is in standard form.

$$\left(x+3\right)A=2-5B+x-Bx$$

Divide both sides by x+3.

$$\frac{\left(x+3\right)A}{x+3}=\frac{2-5B+x-Bx}{x+3}$$

Dividing by x+3 undoes the multiplication by x+3.

$$A=\frac{2-5B+x-Bx}{x+3}$$

Solve for B

$$B=-\frac{Ax-x+3A-2}{x+5}$$

$$x\neq -5\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of \left(x+5\right)\left(x+3\right),x+5,x+3.

$$x+2=\left(x+3\right)A+\left(x+5\right)B$$

Use the distributive property to multiply x+3 by A.

$$x+2=xA+3A+\left(x+5\right)B$$

Use the distributive property to multiply x+5 by B.

$$x+2=xA+3A+xB+5B$$

Swap sides so that all variable terms are on the left hand side.

$$xA+3A+xB+5B=x+2$$

Subtract xA from both sides.

$$3A+xB+5B=x+2-xA$$

Subtract 3A from both sides.

$$xB+5B=x+2-xA-3A$$

Combine all terms containing B.

$$\left(x+5\right)B=x+2-xA-3A$$

The equation is in standard form.

$$\left(x+5\right)B=2-3A+x-Ax$$

Divide both sides by x+5.

$$\frac{\left(x+5\right)B}{x+5}=\frac{2-3A+x-Ax}{x+5}$$

Dividing by x+5 undoes the multiplication by x+5.

$$B=\frac{2-3A+x-Ax}{x+5}$$