Question

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

Answer 1

Hint: In this question we have given two pairs of coordinates for which we need to find an equation of line that contains both the points. In order to do so first we need to determine the slope of the line. We can use the following formula to find the slope: \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] , where m is the slope and \[\left( {{x_1},{y_1}} \right)\] and \[({x_2},{y_2})\] are the two points on the line.

Complete step-by-step solution:
We will use the following formula to find the slope:
\[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Substituting the values from the points in the problem gives the value of slope as such:
$\Rightarrow$\[m = \dfrac{{3 - 1}}{{9 - 3}} = \dfrac{2}{6} = \dfrac{1}{3}\]
Now, we will use the point-slope form of the equation to find the required equation of the line. The point-slope formula states: \[\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)\]
Where m is the slope and \[\left( {{x_1},{y_1}} \right)\] is a point on the line that passes through it.
Substituting the slope in the line equation that we calculated above and the first point from the problem, we get,
$\Rightarrow$\[\left( {y - 1} \right) = \dfrac{1}{3}\left( {x - 3} \right)\]
We can also substitute the slope we calculated and the second point from the problem giving:
$\Rightarrow$\[\left( {y - 3} \right) = \dfrac{1}{3}\left( {x - 9} \right)\]
Simplifying the equation,
$\Rightarrow$\[3\left( {y - 3} \right) = \left( {x - 9} \right)\]
$\Rightarrow$$3y - 9 = x - 9$

Hence, our final equation is-
$3y - x = - 9 + 9 = 0$
$3y - x = 0$

Note: We can also solve from the formula of slope-intercept form. The slope-intercept form of a linear equation is:
\[y = mx + b\]
Where m is the slope and b is the y-intercept value.
Solving our second equation for y gives:
$\Rightarrow$\[y - 3 = \left( {\dfrac{1}{3} \times x} \right) - \left( {\dfrac{1}{3} \times 9} \right)\]
Further solving above equation we get;
$\Rightarrow$\[y - 3 = \dfrac{1}{3}x - 3\]
On adding 3 on both sides of the equation we arrive at solution as such:
$\Rightarrow$\[y - 3 + 3 = \dfrac{1}{3}x - 3 + 3\]
Thus, our final equation is as follows: \[y = \dfrac{1}{3}x + 0\;or\;y = \dfrac{1}{3}x\]
Hence the four equations which solve this problem are:
$\Rightarrow$\[\left( {y - 1} \right) = \dfrac{1}{3}\left( {x - 3} \right)\;or\]
$\Rightarrow$\[\left( {y - 3} \right) = \dfrac{1}{3}\left( {x - 9} \right)\;or\]
$\Rightarrow$\[y = \dfrac{1}{3}x + 0\;or\;y = \dfrac{1}{3}x\]
Remember, changing the form of a line's equation does not change the line. It simply rewrites the variables in a different way. Also, one can simply use the differences in the \[y\] and \[x\] values to find the slope then substitute the values of one point into the equation to find b the \[y\] intercept.