Solve f(x)=3x-1yg(x)=x^2-x-2quad(f-g)(x)

$$f ( x ) = 3 x - 1 y g ( x ) = x ^ { 2 } - x - 2 \quad ( f - g ) ( x )$$

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=\frac{x+gy+2g-4}{2}\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}-x-2\left(f-g\right)x=3x-ygx$$

Use the distributive property to multiply -2 by f-g.

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

Use the distributive property to multiply -2f+2g by x.

$$x^{2}-x-2fx+2gx=3x-ygx$$

Subtract x^{2} from both sides.

$$-x-2fx+2gx=3x-ygx-x^{2}$$

Add x to both sides.

$$-2fx+2gx=3x-ygx-x^{2}+x$$

Combine 3x and x to get 4x.

$$-2fx+2gx=4x-ygx-x^{2}$$

Subtract 2gx from both sides.

$$-2fx=4x-ygx-x^{2}-2gx$$

The equation is in standard form.

$$\left(-2x\right)f=4x-2gx-gxy-x^{2}$$

Divide both sides by -2x.

$$\frac{\left(-2x\right)f}{-2x}=\frac{x\left(4-2g-gy-x\right)}{-2x}$$

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

$$f=\frac{x\left(4-2g-gy-x\right)}{-2x}$$

Divide x\left(4-yg-x-2g\right) by -2x.

$$f=\frac{gy}{2}+\frac{x}{2}+g-2$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{4+2f-x}{y+2}\text{, }&y\neq -2\\g\in \mathrm{C}\text{, }&x=0\text{ or }\left(x=2f+4\text{ and }y=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 2\left(f-g\right)x to both sides.

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

Use the distributive property to multiply 2 by f-g.

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

Use the distributive property to multiply 2f-2g by x.

$$3x-ygx+2fx-2gx=x^{2}-x$$

Subtract 3x from both sides.

$$-ygx+2fx-2gx=x^{2}-x-3x$$

Combine -x and -3x to get -4x.

$$-ygx+2fx-2gx=x^{2}-4x$$

Subtract 2fx from both sides.

$$-ygx-2gx=x^{2}-4x-2fx$$

Combine all terms containing g.

$$\left(-yx-2x\right)g=x^{2}-4x-2fx$$

The equation is in standard form.

$$\left(-xy-2x\right)g=x^{2}-2fx-4x$$

Divide both sides by -yx-2x.

$$\frac{\left(-xy-2x\right)g}{-xy-2x}=\frac{x\left(x-2f-4\right)}{-xy-2x}$$

Dividing by -yx-2x undoes the multiplication by -yx-2x.

$$g=\frac{x\left(x-2f-4\right)}{-xy-2x}$$

Divide x\left(-4+x-2f\right) by -yx-2x.

$$g=-\frac{x-2f-4}{y+2}$$

Solve for f

$$\left\{\begin{matrix}\\f=\frac{x+gy+2g-4}{2}\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}-x-2\left(f-g\right)x=3x-ygx$$

Use the distributive property to multiply -2 by f-g.

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

Use the distributive property to multiply -2f+2g by x.

$$x^{2}-x-2fx+2gx=3x-ygx$$

Subtract x^{2} from both sides.

$$-x-2fx+2gx=3x-ygx-x^{2}$$

Add x to both sides.

$$-2fx+2gx=3x-ygx-x^{2}+x$$

Combine 3x and x to get 4x.

$$-2fx+2gx=4x-ygx-x^{2}$$

Subtract 2gx from both sides.

$$-2fx=4x-ygx-x^{2}-2gx$$

The equation is in standard form.

$$\left(-2x\right)f=4x-2gx-gxy-x^{2}$$

Divide both sides by -2x.

$$\frac{\left(-2x\right)f}{-2x}=\frac{x\left(4-2g-gy-x\right)}{-2x}$$

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

$$f=\frac{x\left(4-2g-gy-x\right)}{-2x}$$

Divide x\left(4-yg-x-2g\right) by -2x.

$$f=\frac{gy}{2}+\frac{x}{2}+g-2$$

Solve for g

$$\left\{\begin{matrix}g=\frac{4+2f-x}{y+2}\text{, }&y\neq -2\\g\in \mathrm{R}\text{, }&x=0\text{ or }\left(x=2f+4\text{ and }y=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 2\left(f-g\right)x to both sides.

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

Use the distributive property to multiply 2 by f-g.

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

Use the distributive property to multiply 2f-2g by x.

$$3x-ygx+2fx-2gx=x^{2}-x$$

Subtract 3x from both sides.

$$-ygx+2fx-2gx=x^{2}-x-3x$$

Combine -x and -3x to get -4x.

$$-ygx+2fx-2gx=x^{2}-4x$$

Subtract 2fx from both sides.

$$-ygx-2gx=x^{2}-4x-2fx$$

Combine all terms containing g.

$$\left(-yx-2x\right)g=x^{2}-4x-2fx$$

The equation is in standard form.

$$\left(-xy-2x\right)g=x^{2}-2fx-4x$$

Divide both sides by -yx-2x.

$$\frac{\left(-xy-2x\right)g}{-xy-2x}=\frac{x\left(x-2f-4\right)}{-xy-2x}$$

Dividing by -yx-2x undoes the multiplication by -yx-2x.

$$g=\frac{x\left(x-2f-4\right)}{-xy-2x}$$

Divide x\left(-4+x-2f\right) by -yx-2x.

$$g=-\frac{x-2f-4}{y+2}$$