Solve Y_{1}=beta_{1}+beta_{2}Y_{2}+beta_{3}X_{1}+varepsilon

$$Y _ { 1 } = \beta _ { 1 } + \beta _ { 2 } Y _ { 2 } + \beta _ { 3 } X _ { 1 } + \varepsilon$$

Solve for X_1

$$\left\{\begin{matrix}X_{1}=-\frac{\epsilon +\beta _{1}+Y_{2}\beta _{2}-Y_{1}}{\beta _{3}}\text{, }&\beta _{3}\neq 0\\X_{1}\in \mathrm{R}\text{, }&Y_{1}=Y_{2}\beta _{2}+\beta _{1}+\epsilon \text{ and }\beta _{3}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\beta _{1}+\beta _{2}Y_{2}+\beta _{3}X_{1}+\epsilon =Y_{1}$$

Subtract \beta _{1} from both sides.

$$\beta _{2}Y_{2}+\beta _{3}X_{1}+\epsilon =Y_{1}-\beta _{1}$$

Subtract \beta _{2}Y_{2} from both sides.

$$\beta _{3}X_{1}+\epsilon =Y_{1}-\beta _{1}-\beta _{2}Y_{2}$$

Subtract \epsilon from both sides.

$$\beta _{3}X_{1}=Y_{1}-\beta _{1}-\beta _{2}Y_{2}-\epsilon$$

The equation is in standard form.

$$\beta _{3}X_{1}=Y_{1}-Y_{2}\beta _{2}-\beta _{1}-\epsilon$$

Divide both sides by \beta _{3}.

$$\frac{\beta _{3}X_{1}}{\beta _{3}}=\frac{Y_{1}-Y_{2}\beta _{2}-\beta _{1}-\epsilon }{\beta _{3}}$$

Dividing by \beta _{3} undoes the multiplication by \beta _{3}.

$$X_{1}=\frac{Y_{1}-Y_{2}\beta _{2}-\beta _{1}-\epsilon }{\beta _{3}}$$

Solve for Y_1

$$Y_{1}=X_{1}\beta _{3}+Y_{2}\beta _{2}+\beta _{1}+\epsilon$$