Solve y-y_{1}=frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})

$$y - y _ { 1 } = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } } ( x - x _ { 1 } )$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{x_{1}y-x_{1}y_{2}-x_{2}y+x_{2}y_{1}}{y_{2}-y_{1}}\text{, }&x_{2}\neq x_{1}\text{ and }y_{1}\neq y_{2}\\x\in \mathrm{R}\text{, }&y_{1}=y_{2}\text{ and }x_{2}\neq x_{1}\text{ and }y=y_{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by -x_{1}+x_{2}.

$$\left(-x_{1}+x_{2}\right)y-y_{1}\left(-x_{1}+x_{2}\right)=\left(y_{2}-y_{1}\right)\left(x-x_{1}\right)$$

Use the distributive property to multiply -x_{1}+x_{2} by y.

$$-x_{1}y+x_{2}y-y_{1}\left(-x_{1}+x_{2}\right)=\left(y_{2}-y_{1}\right)\left(x-x_{1}\right)$$

Use the distributive property to multiply -y_{1} by -x_{1}+x_{2}.

$$-x_{1}y+x_{2}y+y_{1}x_{1}-y_{1}x_{2}=\left(y_{2}-y_{1}\right)\left(x-x_{1}\right)$$

Use the distributive property to multiply y_{2}-y_{1} by x-x_{1}.

$$-x_{1}y+x_{2}y+y_{1}x_{1}-y_{1}x_{2}=y_{2}x-y_{2}x_{1}-y_{1}x+y_{1}x_{1}$$

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

$$y_{2}x-y_{2}x_{1}-y_{1}x+y_{1}x_{1}=-x_{1}y+x_{2}y+y_{1}x_{1}-y_{1}x_{2}$$

Add y_{2}x_{1} to both sides.

$$y_{2}x-y_{1}x+y_{1}x_{1}=-x_{1}y+x_{2}y+y_{1}x_{1}-y_{1}x_{2}+y_{2}x_{1}$$

Subtract y_{1}x_{1} from both sides.

$$y_{2}x-y_{1}x=-x_{1}y+x_{2}y+y_{1}x_{1}-y_{1}x_{2}+y_{2}x_{1}-y_{1}x_{1}$$

Combine y_{1}x_{1} and -y_{1}x_{1} to get 0.

$$y_{2}x-y_{1}x=-x_{1}y+x_{2}y-y_{1}x_{2}+y_{2}x_{1}$$

Combine all terms containing x.

$$\left(y_{2}-y_{1}\right)x=-x_{1}y+x_{2}y-y_{1}x_{2}+y_{2}x_{1}$$

The equation is in standard form.

$$\left(y_{2}-y_{1}\right)x=-x_{1}y+x_{1}y_{2}+x_{2}y-x_{2}y_{1}$$

Divide both sides by y_{2}-y_{1}.

$$\frac{\left(y_{2}-y_{1}\right)x}{y_{2}-y_{1}}=\frac{-x_{1}y+x_{1}y_{2}+x_{2}y-x_{2}y_{1}}{y_{2}-y_{1}}$$

Dividing by y_{2}-y_{1} undoes the multiplication by y_{2}-y_{1}.

$$x=\frac{-x_{1}y+x_{1}y_{2}+x_{2}y-x_{2}y_{1}}{y_{2}-y_{1}}$$

Solve for x_1

$$\left\{\begin{matrix}x_{1}=-\frac{xy_{1}-xy_{2}+x_{2}y-x_{2}y_{1}}{y_{2}-y}\text{, }&x_{2}\neq x\text{ and }y_{1}\neq y_{2}\text{ and }y\neq y_{2}\\x_{1}\neq x_{2}\text{, }&\left(y=y_{2}\text{ and }y_{1}=y_{2}\right)\text{ or }\left(y\neq y_{1}\text{ and }y_{2}=y\text{ and }x=x_{2}\right)\end{matrix}\right.$$