Solve Y_{2}=gamma_{1}+gamma_{2}Y_{1}+gamma_{2}X_{1}+gamma_{4}X_{2}+u

$$Y _ { 2 } = \gamma _ { 1 } + \gamma _ { 2 } Y _ { 1 } + \gamma _ { 2 } X _ { 1 } + \gamma _ { 4 } X _ { 2 } + u$$

Solve for X_1

$$\left\{\begin{matrix}X_{1}=-\frac{u+X_{2}\gamma _{4}+Y_{1}\gamma _{2}+\gamma _{1}-Y_{2}}{\gamma _{2}}\text{, }&\gamma _{2}\neq 0\\X_{1}\in \mathrm{R}\text{, }&Y_{2}=u+X_{2}\gamma _{4}+\gamma _{1}\text{ and }\gamma _{2}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\gamma _{1}+\gamma _{2}Y_{1}+\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}$$

Subtract \gamma _{1} from both sides.

$$\gamma _{2}Y_{1}+\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}-\gamma _{1}$$

Subtract \gamma _{2}Y_{1} from both sides.

$$\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}$$

Subtract \gamma _{4}X_{2} from both sides.

$$\gamma _{2}X_{1}+u=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}-\gamma _{4}X_{2}$$

Subtract u from both sides.

$$\gamma _{2}X_{1}=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}-\gamma _{4}X_{2}-u$$

The equation is in standard form.

$$\gamma _{2}X_{1}=-u-X_{2}\gamma _{4}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}$$

Divide both sides by \gamma _{2}.

$$\frac{\gamma _{2}X_{1}}{\gamma _{2}}=\frac{-u-X_{2}\gamma _{4}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}}{\gamma _{2}}$$

Dividing by \gamma _{2} undoes the multiplication by \gamma _{2}.

$$X_{1}=\frac{-u-X_{2}\gamma _{4}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}}{\gamma _{2}}$$

Solve for X_2

$$\left\{\begin{matrix}X_{2}=-\frac{u+X_{1}\gamma _{2}+Y_{1}\gamma _{2}+\gamma _{1}-Y_{2}}{\gamma _{4}}\text{, }&\gamma _{4}\neq 0\\X_{2}\in \mathrm{R}\text{, }&u=-X_{1}\gamma _{2}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}\text{ and }\gamma _{4}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\gamma _{1}+\gamma _{2}Y_{1}+\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}$$

Subtract \gamma _{1} from both sides.

$$\gamma _{2}Y_{1}+\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}-\gamma _{1}$$

Subtract \gamma _{2}Y_{1} from both sides.

$$\gamma _{2}X_{1}+\gamma _{4}X_{2}+u=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}$$

Subtract \gamma _{2}X_{1} from both sides.

$$\gamma _{4}X_{2}+u=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}-\gamma _{2}X_{1}$$

Subtract u from both sides.

$$\gamma _{4}X_{2}=Y_{2}-\gamma _{1}-\gamma _{2}Y_{1}-\gamma _{2}X_{1}-u$$

The equation is in standard form.

$$\gamma _{4}X_{2}=-u-X_{1}\gamma _{2}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}$$

Divide both sides by \gamma _{4}.

$$\frac{\gamma _{4}X_{2}}{\gamma _{4}}=\frac{-u-X_{1}\gamma _{2}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}}{\gamma _{4}}$$

Dividing by \gamma _{4} undoes the multiplication by \gamma _{4}.

$$X_{2}=\frac{-u-X_{1}\gamma _{2}-Y_{1}\gamma _{2}+Y_{2}-\gamma _{1}}{\gamma _{4}}$$