Question

How do you write the slope-intercept form of the line with $\left( { - 5,0} \right)$ and $\left( {8, - 4} \right)$?

Answer 1

Here, we are required to write the slope-intercept form of a line which is passing through two given points. Thus, we will write the standard slope-intercept equation of a line and using the formula of slope of a line, we will be able to find the slope of the given line and further substituting one of the given points, we will be able to find the $y$-intercept and hence, we will able to find the required slope-intercept form of the given line.

Formula Used:
1. Slope of a line, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
2. General slope-intercept form of a line is given by: $y = mx + c$,
Where, $m$is the slope and $c$is the $y$-intercept value.

Complete step by step solution:
According to the question, the given points are:
$\left( {{x_1},{y_1}} \right) = \left( { - 5,0} \right)$ and $\left( {{x_2},{y_2}} \right) = \left( {8, - 4} \right)$
Now, when two points on a line are given to us, then in order to find its slope-intercept form, first of all we will find the slope of the line.
Thus, we will use the formula:
Slope, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Thus, substituting $\left( {{x_1},{y_1}} \right) = \left( { - 5,0} \right)$ and $\left( {{x_2},{y_2}} \right) = \left( {8, - 4} \right)$ in this formula, we get,
Slope, $m = \dfrac{{ - 4 - 0}}{{8 - \left( { - 5} \right)}} = \dfrac{{ - 4 - 0}}{{8 + 5}} = \dfrac{{ - 4}}{{13}}$
Now, in order to find the slope-intercept form, we should know that the general slope-intercept form of a line is given by: $y = mx + c$
Where, $m$is the slope and $c$is the $y$-intercept value.
Thus, substituting the value of slope, $m = \dfrac{{ - 4}}{{13}}$, we get,
$y = - \dfrac{4}{{13}}x + c$…………………………$\left( 1 \right)$
Now, in order to find the value of $y$-intercept, $c$, we will substitute any of the given points in this equation.
Thus, substituting $\left( {x,y} \right) = \left( { - 5,0} \right)$, we get,
$0 = - \dfrac{4}{{13}}\left( { - 5} \right) + c$
$\Rightarrow 0 = \dfrac{{20}}{{13}} + c$
Subtracting $\dfrac{{20}}{{13}}$from both the sides, we get,
$\Rightarrow c = - \dfrac{{20}}{{13}}$
Thus, substituting this value of $y$-intercept in $\left( 1 \right)$, we get,
$y = - \dfrac{4}{{13}}x - \dfrac{{20}}{{13}}$
Therefore, clearly, this is in the slope-intercept form $y = mx + c$

Hence, the slope-intercept form of the line passing through the points $\left( { - 5,0} \right)$ and $\left( {8, - 4} \right)$ is $y = - \dfrac{4}{{13}}x - \dfrac{{20}}{{13}}$
Thus, this is the required answer.

Note:
In geometry, a line can be defined as a straight one-dimensional figure that has no thickness and extends endlessly in both directions. It is sometimes described as the shortest distance between any two points. Whereas, a line segment is only a part of a line and it has two endpoints.
The standard form for linear equations in two variables or a line passing through two points is $Ax + By = C$. When an equation is given in this form then we can say that it is in the standard form, whereas, the slope-intercept form: $y = mx + c$ emphasizes on the slope, $m$ and the $y$-intercept of the line.