Solve A/x-1-B/x+2=x+5/(x-1)(x+2)

$$\frac { A } { x - 1 } - \frac { B } { x + 2 } = \frac { x + 5 } { ( x - 1 ) ( x + 2 ) }$$

Solve for A

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

$$x\neq 1\text{ and }x\neq -2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,\left(x-1\right)\left(x+2\right).

$$\left(x+2\right)A-\left(x-1\right)B=x+5$$

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

$$xA+2A-\left(x-1\right)B=x+5$$

Use the distributive property to multiply x-1 by B.

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

To find the opposite of xB-B, find the opposite of each term.

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

Add xB to both sides.

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

Subtract B from both sides.

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

Combine all terms containing A.

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

The equation is in standard form.

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

Divide both sides by x+2.

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

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

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

Solve for B

$$B=-\frac{Ax-x+2A-5}{1-x}$$

$$x\neq 1\text{ and }x\neq -2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2,\left(x-1\right)\left(x+2\right).

$$\left(x+2\right)A-\left(x-1\right)B=x+5$$

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

$$xA+2A-\left(x-1\right)B=x+5$$

Use the distributive property to multiply x-1 by B.

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

To find the opposite of xB-B, find the opposite of each term.

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

Subtract xA from both sides.

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

Subtract 2A from both sides.

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

Combine all terms containing B.

$$\left(-x+1\right)B=x+5-xA-2A$$

The equation is in standard form.

$$\left(1-x\right)B=5-2A+x-Ax$$

Divide both sides by -x+1.

$$\frac{\left(1-x\right)B}{1-x}=\frac{5-2A+x-Ax}{1-x}$$

Dividing by -x+1 undoes the multiplication by -x+1.

$$B=\frac{5-2A+x-Ax}{1-x}$$