Solve f(x)+f(x-1/7x+2)=x

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

Solve for f

$$f=\frac{x\left(7x+2\right)}{7x^{2}+3x-1}$$

$$x\neq \frac{\sqrt{37}-3}{14}\text{ and }x\neq \frac{-\sqrt{37}-3}{14}\text{ and }x\neq -\frac{2}{7}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 7x+2.

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

Use the distributive property to multiply fx by 7x+2.

$$7fx^{2}+2fx+f\left(x-1\right)=x\left(7x+2\right)$$

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

$$7fx^{2}+2fx+fx-f=x\left(7x+2\right)$$

Combine 2fx and fx to get 3fx.

$$7fx^{2}+3fx-f=x\left(7x+2\right)$$

Use the distributive property to multiply x by 7x+2.

$$7fx^{2}+3fx-f=7x^{2}+2x$$

Combine all terms containing f.

$$\left(7x^{2}+3x-1\right)f=7x^{2}+2x$$

Divide both sides by 7x^{2}+3x-1.

$$\frac{\left(7x^{2}+3x-1\right)f}{7x^{2}+3x-1}=\frac{x\left(7x+2\right)}{7x^{2}+3x-1}$$

Dividing by 7x^{2}+3x-1 undoes the multiplication by 7x^{2}+3x-1.

$$f=\frac{x\left(7x+2\right)}{7x^{2}+3x-1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{37f^{2}-40f+4}-3f+2}{14\left(f-1\right)}\text{, }&f\neq 1\\x=\frac{\sqrt{37f^{2}-40f+4}-3f+2}{14\left(f-1\right)}\text{, }&f\neq 0\text{ and }f\neq 1\\x=1\text{, }&f=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{37f^{2}-40f+4}-3f+2}{14\left(f-1\right)}\text{, }&f\leq \frac{20-6\sqrt{7}}{37}\text{ or }\left(f\neq 1\text{ and }f\geq \frac{6\sqrt{7}+20}{37}\right)\\x=\frac{\sqrt{37f^{2}-40f+4}-3f+2}{14\left(f-1\right)}\text{, }&\left(f\neq 0\text{ and }f\leq \frac{20-6\sqrt{7}}{37}\right)\text{ or }\left(f\neq 1\text{ and }f\geq \frac{6\sqrt{7}+20}{37}\right)\\x=1\text{, }&f=1\end{matrix}\right.$$