Question

How do you write an equation in standard form of the lines passing through $(2,1)(4,5)$?

Answer 1

Finding the equation of a line passing through two points:

Given points are $(2,1),(4,5)$

The formula for a line passing through given two points $(x_1,y_1)$ and $(x_2,y_2)$ is

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

Put $x_1=2$, $y_1=1$, $x_2=4$ and $y_2=5$

$\Rightarrow$ $y-1=\frac{5-1}{4-2}(x-2)$

$\Rightarrow$ $y-1=\frac{4}{2}(x-2)$

$\Rightarrow$ $y-1=2(x-2)$

$\Rightarrow$ $y-1=2x-4$

$\Rightarrow$ $2x-y=3$

Hence, the equation of a line in standard form is $2x-y=3$.