Solve f^-1x=2x+7

$${ f }^{ -1 } x = 2x+7$$

Solve for f

$$f=\frac{x}{2x+7}$$

$$x\neq 0\text{ and }x\neq -\frac{7}{2}$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}x=2x+7$$

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

$$1x=2xf+f\times 7$$

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

$$2xf+f\times 7=1x$$

Reorder the terms.

$$2fx+7f=x$$

Combine all terms containing f.

$$\left(2x+7\right)f=x$$

Divide both sides by 2x+7.

$$\frac{\left(2x+7\right)f}{2x+7}=\frac{x}{2x+7}$$

Dividing by 2x+7 undoes the multiplication by 2x+7.

$$f=\frac{x}{2x+7}$$

Variable f cannot be equal to 0.

$$f=\frac{x}{2x+7}\text{, }f\neq 0$$

Solve for x

$$x=\frac{7f}{1-2f}$$

$$f\neq \frac{1}{2}\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Subtract 2x from both sides.

$$f^{-1}x-2x=7$$

Reorder the terms.

$$-2x+\frac{1}{f}x=7$$

Multiply both sides of the equation by f.

$$-2xf+1x=7f$$

Reorder the terms.

$$-2fx+x=7f$$

Combine all terms containing x.

$$\left(-2f+1\right)x=7f$$

The equation is in standard form.

$$\left(1-2f\right)x=7f$$

Divide both sides by 1-2f.

$$\frac{\left(1-2f\right)x}{1-2f}=\frac{7f}{1-2f}$$

Dividing by 1-2f undoes the multiplication by 1-2f.

$$x=\frac{7f}{1-2f}$$