Question

How do you find the equation of the line containing the given pair of points $\;\left( {3,1} \right)$ and $\left( {9,3} \right)$?

Answer 1

The equation of a straight line in slope-intercept form is: $y = mx + b$. Where m is the value of slope and b is the y-intercept. Here, m and b are constants, and x and y are variables. Since x and y are variables that describe the position of specific points on the graph, m and b describe features of the function. A straight line is a linear equation of the first order. The slope of a line is the ratio of change in y over the change in x between any two points on the line.
$slope\left( m \right) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Steps to follow:

Find the slope of the line.
Use the slope to find the y-intercept.
Substitute the value of slope and y-intercept in a straight-line equation.

Complete step-by-step answer:
Here, we want to find a line equation. For that two points are given.
Let us compare points $\;\left( {3,1} \right)$ and $\left( {9,3} \right)$ with $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$
Therefore, ${x_1} = 3,{y_1} = 1$ and ${x_2} = 9,{y_2} = 3$.
Now, the first step is to find the slope.
$\Rightarrow slope\left( m \right) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Let us substitute all the values.
$\Rightarrow m = \dfrac{{3 - 1}}{{9 - 3}}$
Subtract it.
$\Rightarrow m = \dfrac{2}{6}$
Let us take out 2 as a common factor.
$\Rightarrow m = \dfrac{1}{3}$
Now, the second step is to find the y-intercept.
For that, we will use a line formula. And substitute the value of ‘m’ in that formula.
$\Rightarrow y = mx + b$
Here, the value of ‘m’ is equal to $\dfrac{1}{3}$.
$\Rightarrow y = \dfrac{1}{3}x + b$...(1)
Now, select any one point to put the value of x and y among the given two points. It does not matter which one of the two points we select because we will get the same answer in either case.
Let us select the first point that is $\;\left( {3,1} \right)$.
Here, the value of x is 3 and the value of y is 1.
Substitute all the values in equation (1).
So,
$\Rightarrow 1 = \dfrac{1}{3} \times 3 + b$
Apply division into the right-hand side.
$\Rightarrow 1 = 1 + b$
Now, let us subtract 1 on both sides.
$\Rightarrow 1 - 1 = 1 - 1 + b$
So,
$\Rightarrow b = 0$
Now, put the value of b in equation (1).
So we get,
$\Rightarrow y = \dfrac{1}{3}x + 0$
Therefore,
$\Rightarrow y = \dfrac{1}{3}x$

Hence, the equation of the line is $y = \dfrac{1}{3}x$.

Note:
Let us find the value of y-intercept by selecting the second point that is $\left( {9,3} \right)$.
Here, the value of x is 9 and the value of y is 3.
Substitute all the values in equation (1).
So,
$\Rightarrow 3 = \dfrac{1}{9} \times 3 + b$
Apply division into the right-hand side.
$\Rightarrow 3 = 3 + b$
Now, let us subtract 3 on both sides.
$\Rightarrow 3 - 3 = 3 - 3 + b$
So,$\Rightarrow b = 0$