Solve z=f(x)+e^4yg(x)

$$z = f ( x ) + e ^ { 4 } y g ( x )$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-e^{4}gy+\frac{z}{x}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&z=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fx+e^{4}ygx=z$$

Subtract e^{4}ygx from both sides.

$$fx=z-e^{4}ygx$$

Reorder the terms.

$$fx=-e^{4}gxy+z$$

The equation is in standard form.

$$xf=z-e^{4}gxy$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{z-e^{4}gxy}{x}$$

Dividing by x undoes the multiplication by x.

$$f=\frac{z-e^{4}gxy}{x}$$

Divide z-e^{4}ygx by x.

$$f=-e^{4}gy+\frac{z}{x}$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\frac{fx-z}{e^{4}xy}\text{, }&y\neq 0\text{ and }x\neq 0\\g\in \mathrm{C}\text{, }&\left(z=0\text{ and }x=0\right)\text{ or }\left(z=fx\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fx+e^{4}ygx=z$$

Subtract fx from both sides.

$$e^{4}ygx=z-fx$$

The equation is in standard form.

$$e^{4}xyg=z-fx$$

Divide both sides by e^{4}yx.

$$\frac{e^{4}xyg}{e^{4}xy}=\frac{z-fx}{e^{4}xy}$$

Dividing by e^{4}yx undoes the multiplication by e^{4}yx.

$$g=\frac{z-fx}{e^{4}xy}$$

Solve for f

$$\left\{\begin{matrix}f=-e^{4}gy+\frac{z}{x}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&z=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fx+e^{4}ygx=z$$

Subtract e^{4}ygx from both sides.

$$fx=z-e^{4}ygx$$

Reorder the terms.

$$fx=-e^{4}gxy+z$$

The equation is in standard form.

$$xf=z-e^{4}gxy$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{z-e^{4}gxy}{x}$$

Dividing by x undoes the multiplication by x.

$$f=\frac{z-e^{4}gxy}{x}$$

Divide z-e^{4}ygx by x.

$$f=-e^{4}gy+\frac{z}{x}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{fx-z}{e^{4}xy}\text{, }&y\neq 0\text{ and }x\neq 0\\g\in \mathrm{R}\text{, }&\left(z=0\text{ and }x=0\right)\text{ or }\left(z=fx\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fx+e^{4}ygx=z$$

Subtract fx from both sides.

$$e^{4}ygx=z-fx$$

The equation is in standard form.

$$e^{4}xyg=z-fx$$

Divide both sides by e^{4}yx.

$$\frac{e^{4}xyg}{e^{4}xy}=\frac{z-fx}{e^{4}xy}$$

Dividing by e^{4}yx undoes the multiplication by e^{4}yx.

$$g=\frac{z-fx}{e^{4}xy}$$