Question

How do you find the point-slope form of the equation of the line passing through the points $(-7, 0)$ and $(5, 4)$?

Answer 1

Hint: To find the point-slope form consider the equation \[y - y_1 = m\left( {x - x_1} \right)\] in which the values of $x_1, y_1$ and $x_2, y_2$ are given. We need to find the value of m i.e., slope of a line by \[m = \dfrac{{y_2 - y_2}}{{x_2 - x_1}}\] in which all the points are known to us.

Formula used: \[y - y_1 = m\left( {x - x_1} \right)\]
In this,
$x_1$ = x-coordinate of the point
$y_1$ = y-coordinate of the point
m is the slope.

Complete step-by-step solution:
Let us write the given points as
$(x_1, y_1)$ = (-7, 0)
$(x_2, y_2)$ = (5, 4)
Using point slope form of a line as
\[\Rightarrow y - y_1 = m\left( {x - x_1} \right)\] ……………… 1
In which m is the slope and $x_1, y_1$ are the given points on the line. Hence, we need to determine the value of m i.e., slope given by
\[\Rightarrow m = \dfrac{{y_2 - y_2}}{{x_2 - x_1}}\]
Substitute the values of each terms as
\[\Rightarrow m = \dfrac{{4 - 0}}{{5 - \left( { - 7} \right)}}\]
\[\Rightarrow m = \dfrac{4}{{12}}\]
Hence, after simplifying the slope is
\[\Rightarrow m = \dfrac{1}{3}\]
Now let us use equation 1 i.e., point slope form, in the equation we can use any values either $x_1, y_1$ or $x_2, y_2$.
Let us consider $x_2, y_2$ as
\[\Rightarrow y - y_2 = m\left( {x - x_2} \right)\]
Let us substitute the values of slope m, $x_2$ and $y_2$ we get
\[\Rightarrow y - 4 = \dfrac{1}{3}\left( {x - 5} \right)\]

Therefore, the point-slope form of the equation of line passing through the points is \[y - 4 = \dfrac{1}{3}\left( {x - 5} \right)\].

Note: When the equation of a line using the slope of the line and a point through which the line passes, that equation can be found using the point-slope formula. The equation of a line whose slope is m and which passes through a point $(x_1, y_1)$ is found using the point-slope form and is given as \[y - y_1 = m\left( {x - x_1} \right)\].