How to find the equation of a line passing through two points (Two-point form )

We are going to find the equation of line passing through two given points (\(x_1,y_1\)) and (\(x_2,y_2\)),this is also called two-point form .We can write the equation of line by solving following formula. \[ \frac{y_2-y_1}{x_2-x_1}=\frac{y-y_1}{x-x_1}\] . Lets try an example , suppose we want to find the equation of line through points (\(6,2\)) and (\(10,0\)) ,we will use above formula ,here

\[(x_1,y_1)=(6,2) \;\;\textbf{and }\;\; (x_2,y_2)=(10,0)\]

so we have  \[x_1=6 ,\;\;x_2=10 \;\; \textbf{and}  \;\;y_1=2  \;\;,y_2=0\]

now put these values in above formula

\[ \frac{0-2}{10-6}=\frac{y-2}{x-6}\] 

\[-2(x-6)=(y-2)(4)\]

\[\Rightarrow -2x +12 =4y -8 \]

\[\Rightarrow -2x=4y-8-12\]

\[\Rightarrow -2x=4y-20\] 

so the equation of line passing through the given points is \(-2x=4y-20\).see the following fig.