Solve f^-1=5x/x-6

$$f ^ { - 1 } = \frac { 5 x } { x - 6 }$$

Solve for f

$$f=\frac{1}{5}-\frac{6}{5x}$$

$$x\neq 6\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-6.

$$\left(x-6\right)f^{-1}=5x$$

Use the distributive property to multiply x-6 by f^{-1}.

$$xf^{-1}-6f^{-1}=5x$$

Reorder the terms.

$$\frac{1}{f}x-6\times \frac{1}{f}=5x$$

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.

$$1x-6=5xf$$

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

$$5xf=1x-6$$

Reorder the terms.

$$5fx=x-6$$

The equation is in standard form.

$$5xf=x-6$$

Divide both sides by 5x.

$$\frac{5xf}{5x}=\frac{x-6}{5x}$$

Dividing by 5x undoes the multiplication by 5x.

$$f=\frac{x-6}{5x}$$

Divide x-6 by 5x.

$$f=\frac{1}{5}-\frac{6}{5x}$$

Variable f cannot be equal to 0.

$$f=\frac{1}{5}-\frac{6}{5x}\text{, }f\neq 0$$

Solve for x

$$x=-\frac{6}{5f-1}$$

$$f\neq 0\text{ and }f\neq \frac{1}{5}$$

Steps for Solving Linear Equation

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

$$\left(x-6\right)f^{-1}=5x$$

Use the distributive property to multiply x-6 by f^{-1}.

$$xf^{-1}-6f^{-1}=5x$$

Subtract 5x from both sides.

$$xf^{-1}-6f^{-1}-5x=0$$

Add 6f^{-1} to both sides. Anything plus zero gives itself.

$$xf^{-1}-5x=6f^{-1}$$

Reorder the terms.

$$-5x+\frac{1}{f}x=6\times \frac{1}{f}$$

Multiply both sides of the equation by f.

$$-5xf+1x=6\times 1$$

Multiply 6 and 1 to get 6.

$$-5xf+1x=6$$

Reorder the terms.

$$-5fx+x=6$$

Combine all terms containing x.

$$\left(-5f+1\right)x=6$$

The equation is in standard form.

$$\left(1-5f\right)x=6$$

Divide both sides by -5f+1.

$$\frac{\left(1-5f\right)x}{1-5f}=\frac{6}{1-5f}$$

Dividing by -5f+1 undoes the multiplication by -5f+1.

$$x=\frac{6}{1-5f}$$

Variable x cannot be equal to 6.

$$x=\frac{6}{1-5f}\text{, }x\neq 6$$