Answer
The equation of a line passing through two points is given by first calculating the slope of the line and then applying the formula of slope-point form.
Work Step by Step
Consider a line passing through the two given points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$.
Follow the steps given below to find the equation of the line, following the steps below:
Step 1: Calculate slope for the line passing through the distinct points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ using the formula given below
$m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Step 2: Apply the slope-point form to find the equation of the line using the formula given below:
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Example:
Consider a line passing through the given points $\left( 3,-2 \right)$ and $\left( 1,4 \right)$.
Apply step 1:
Calculate the slope for the line passing through the distinct points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ using the formula given below
$m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Substitute $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 3,-2 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 1,4 \right)$; the slope of the required line is:
$\begin{align}
& m=\frac{4-\left( -2 \right)}{1-3} \\
& =\frac{4+2}{\left( -2 \right)} \\
& =\frac{6}{\left( -2 \right)} \\
& =-3
\end{align}$
Thus, the slope of the given line is $m=-3$.
Apply step 2:
Use point–slope form of the equation of a line passing through point $\left( {{x}_{1}},{{y}_{1}} \right)$ and with a slope m:
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Replace $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 3,-2 \right)$ and $m=-3$; the equation of the required line is:
$\begin{align}
& y-\left( -2 \right)=-3\left( x-3 \right) \\
& y+2=-3\left( x-3 \right)
\end{align}$
Thus, the equation of the line is $y+2=-3\left( x-3 \right)$.
