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

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

Solve for A

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

$$x\neq -2\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of \left(x+2\right)\left(x+3\right),x+2,x+3.

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

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

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

Use the distributive property to multiply x+2 by B.

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

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

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

Subtract xB from both sides.

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

Subtract 2B from both sides.

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

Combine all terms containing A.

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

The equation is in standard form.

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

Divide both sides by x+3.

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

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

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

Solve for B

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

$$x\neq -2\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of \left(x+2\right)\left(x+3\right),x+2,x+3.

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

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

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

Use the distributive property to multiply x+2 by B.

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

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

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

Subtract xA from both sides.

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

Subtract 3A from both sides.

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

Combine all terms containing B.

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

The equation is in standard form.

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

Divide both sides by x+2.

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

Dividing by x+2 undoes the multiplication by x+2.

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