Question

How do you write the equation in point slope form given x-intercept of -4 and a y-intercept of -1?

Answer 1

Hint: This problem deals with expressing an equation of a line in point slope form. Point slope is the general form $y - {y_1} = m\left( {x - {x_1}} \right)$ for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). This point slope is useful for finding the equation for a line when you know one point along the line and the slope of the line.

Complete step-by-step solution:
Here given the x-intercept and the y-intercept of a line.
If the x-intercept is $a$, and the y-intercept is $b$, then equation of that line is given by:
$ \Rightarrow \dfrac{x}{a} + \dfrac{y}{b} = 1$
Here the given x-intercept is -4 and the y-intercept is -1, so the equation of the line is :
$ \Rightarrow \dfrac{x}{{ - 4}} + \dfrac{y}{{ - 1}} = 1$
Now simplify the above equation of line to get the slope of the equation, as shown:
\[ \Rightarrow \dfrac{{x + 4y}}{{ - 4}} = 1\]
\[ \Rightarrow x + 4y = - 4\]
This line passes through the point $\left( {0, - 1} \right)$, this point is $\left( {{x_1},{y_1}} \right)$
\[ \Rightarrow 4y = - x - 4\]
\[ \Rightarrow y = - \dfrac{x}{4} - \dfrac{4}{4}\]
Dividing the equation by 4, as shown above.
\[ \Rightarrow y = - \dfrac{1}{4}x - 1\]
This equation is in the form of $y = mx + c$
Here the slope of the equation is $m = - \dfrac{1}{4}$
The y-intercept is $c = - 1$.
The point slope form is given by:
$ \Rightarrow \left( {y - {y_1}} \right) = \dfrac{{ - 1}}{4}\left( {x - {x_1}} \right)$
As considered above here $\left( {{x_1},{y_1}} \right) = \left( {0, - 1} \right)$, or any point on the line.
$ \Rightarrow \left( {y - \left( { - 1} \right)} \right) = \dfrac{{ - 1}}{4}\left( {x - 0} \right)$
$ \Rightarrow \left( {y + 1} \right) = \dfrac{{ - 1}}{4}\left( {x - 0} \right)$

The point slope form is $\left( {y - {y_1}} \right) = \dfrac{{ - 1}}{4}\left( {x - {x_1}} \right)$

Note: Please note that this point slope form is derived from the equation for finding the slope of a line and has practical uses in many areas of mathematics and the real world. The equation of a line is typically written as $y = mx + b$, where $m$ is the slope and $b$ is the intercept.