Solve 1/x^2-5x-6=a/x-6+b/x+1

$$\frac { 1 } { x ^ { 2 } - 5 x - 6 } = \frac { a } { x - 6 } + \frac { b } { x + 1 }$$

Solve for a

$$a=-\frac{bx-6b-1}{x+1}$$

$$x\neq 6\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-6\right)\left(x+1\right), the least common multiple of x^{2}-5x-6,x-6,x+1.

$$1=\left(x+1\right)a+\left(x-6\right)b$$

Use the distributive property to multiply x+1 by a.

$$1=xa+a+\left(x-6\right)b$$

Use the distributive property to multiply x-6 by b.

$$1=xa+a+xb-6b$$

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

$$xa+a+xb-6b=1$$

Subtract xb from both sides.

$$xa+a-6b=1-xb$$

Add 6b to both sides.

$$xa+a=1-xb+6b$$

Combine all terms containing a.

$$\left(x+1\right)a=1-xb+6b$$

The equation is in standard form.

$$\left(x+1\right)a=1+6b-bx$$

Divide both sides by x+1.

$$\frac{\left(x+1\right)a}{x+1}=\frac{1+6b-bx}{x+1}$$

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

$$a=\frac{1+6b-bx}{x+1}$$

Solve for b

$$b=-\frac{ax+a-1}{x-6}$$

$$x\neq 6\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-6\right)\left(x+1\right), the least common multiple of x^{2}-5x-6,x-6,x+1.

$$1=\left(x+1\right)a+\left(x-6\right)b$$

Use the distributive property to multiply x+1 by a.

$$1=xa+a+\left(x-6\right)b$$

Use the distributive property to multiply x-6 by b.

$$1=xa+a+xb-6b$$

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

$$xa+a+xb-6b=1$$

Subtract xa from both sides.

$$a+xb-6b=1-xa$$

Subtract a from both sides.

$$xb-6b=1-xa-a$$

Combine all terms containing b.

$$\left(x-6\right)b=1-xa-a$$

The equation is in standard form.

$$\left(x-6\right)b=1-a-ax$$

Divide both sides by -6+x.

$$\frac{\left(x-6\right)b}{x-6}=\frac{1-a-ax}{x-6}$$

Dividing by -6+x undoes the multiplication by -6+x.

$$b=\frac{1-a-ax}{x-6}$$