Question

How do you find the equation for the line with points $\left( 1,3 \right)$ and $\left( 5,9 \right)$?

Answer 1

Hint: Now we are given with two points on the line. The equation of line passing $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $\dfrac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\dfrac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ . Hence we will use this two point form of the equation to write the required equation of the line. Now we will rearrange the equation in general form which is $ax+by+c=0$ hence we have the required equation of the line.

Complete step by step solution:
Now we are given two points on the line.
Now we know that there exists only one line passing through two points.
Now we want to find the equation of the line passing through two points.
Now we know the general equation of the line is given by $ax+by+c=0$ to the equation of the line we will write the equation in two point form and then arrange it in the general equation.
The equation of line in two point form passing through points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $\dfrac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\dfrac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Now let $\left( 1,3 \right)$ be $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( 5,9 \right)$ be $\left( {{x}_{2}},{{y}_{2}} \right)$ hence using the formula we get,
$\begin{align}
  & \Rightarrow \dfrac{y-3}{9-3}=\dfrac{x-1}{5-1} \\
 & \Rightarrow \dfrac{y-3}{6}=\dfrac{x-1}{4} \\
\end{align}$
Now cross multiplying the equation we get,
$\begin{align}
  & \Rightarrow 4\left( y-3 \right)=6\left( x-1 \right) \\
 & \Rightarrow 4y-12=6x-6 \\
\end{align}$
Now rearranging the equation we get $6x-4y+6=0$
Hence we have the equation of line in general form $ax+by+c=0$ and we have the required equation of the line.

Note: Now note that the c in the formula is called y intercept and is the intersection of line and y axis. Similarly we have x intercept which is the intersection of line and x axis. Now if x intercept is a and y intercept is b, then the equation of the line is given by $\dfrac{x}{a}+\dfrac{y}{b}=c$ .