Solve s+7/(s+5)(s+1)=A/(s+5)+B/(s+1)

$$\frac { s + 7 } { ( s + 5 ) ( s + 1 ) } = \frac { A } { ( s + 5 ) } + \frac { B } { ( s + 1 ) }$$

Solve for A

$$A=-\frac{Bs-s+5B-7}{s+1}$$

$$s\neq -5\text{ and }s\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(s+1\right)\left(s+5\right), the least common multiple of \left(s+5\right)\left(s+1\right),s+5,s+1.

$$s+7=\left(s+1\right)A+\left(s+5\right)B$$

Use the distributive property to multiply s+1 by A.

$$s+7=sA+A+\left(s+5\right)B$$

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

$$s+7=sA+A+sB+5B$$

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

$$sA+A+sB+5B=s+7$$

Subtract sB from both sides.

$$sA+A+5B=s+7-sB$$

Subtract 5B from both sides.

$$sA+A=s+7-sB-5B$$

Combine all terms containing A.

$$\left(s+1\right)A=s+7-sB-5B$$

The equation is in standard form.

$$\left(s+1\right)A=7-5B+s-Bs$$

Divide both sides by s+1.

$$\frac{\left(s+1\right)A}{s+1}=\frac{7-5B+s-Bs}{s+1}$$

Dividing by s+1 undoes the multiplication by s+1.

$$A=\frac{7-5B+s-Bs}{s+1}$$

Solve for B

$$B=-\frac{As-s+A-7}{s+5}$$

$$s\neq -5\text{ and }s\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(s+1\right)\left(s+5\right), the least common multiple of \left(s+5\right)\left(s+1\right),s+5,s+1.

$$s+7=\left(s+1\right)A+\left(s+5\right)B$$

Use the distributive property to multiply s+1 by A.

$$s+7=sA+A+\left(s+5\right)B$$

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

$$s+7=sA+A+sB+5B$$

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

$$sA+A+sB+5B=s+7$$

Subtract sA from both sides.

$$A+sB+5B=s+7-sA$$

Subtract A from both sides.

$$sB+5B=s+7-sA-A$$

Combine all terms containing B.

$$\left(s+5\right)B=s+7-sA-A$$

The equation is in standard form.

$$\left(s+5\right)B=7-A+s-As$$

Divide both sides by s+5.

$$\frac{\left(s+5\right)B}{s+5}=\frac{7-A+s-As}{s+5}$$

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

$$B=\frac{7-A+s-As}{s+5}$$