Question

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

Answer 1

How do you write an equation in standard form of the lines:

The points through which the line passes is given as $(2,1)$ and $(4,5)$.

Let $x_1=2,y_1=1,x_2=4$ and $y_2=5$.Then,

The standard form of the line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is given formula $\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$.

Insert the values of x in the above formula to get the standard equation of the line passing through the points $(2,1)$ and $(4,5)$.

$$\begin{gathered} \frac{y-1}{5-1}=\frac{x-2}{4-2} \\ \Rightarrow \frac{y-1}{4}=\frac{x-2}{2} \\ \Rightarrow 2·(y-1)=4·(x-2) \\ \Rightarrow 2y-2=4x-8 \\ \Rightarrow 4x-2y=8-2 \\ \Rightarrow 4x-2y=6 \\ \Rightarrow 2x-y=3 \end{gathered}$$

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