Question

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

Answer 1

In the above type of question where we are given with the x intercept and the y intercept and also what type of lines to be used for this we will use its general equation and then use the x intercept and the y intercept to find the final equation. As these two lines represent a single line the slope of the line will be the same with which both the lines can be found.

Complete step by step answer:
In the above stated question we have been given two point i.e. the x intercept and the y intercept, these x and y intercept are values when the value of y is zero (0) and the values of x is zero (0) i.e. when the line touches the y-axis the value os x is zero and the value of y that comes will be regarded as y-intercept and when the lines touches the x-axis the value of y in the line equation becomes zero (0) and the value of x at that point will be regarded as x-intercept.
Now we will write the general equation for point slope form and slope intercept form:
Slope-intercept form: $y=mx+b$
Point slope form: $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
From the point slope form we can figure out the slope of line which we can use in slope intercept form and find the equation of line.
now for the point slope form to work we need two point which we have as intercepts so when we see the x-intercept the point x-intercept will have will be (-4,0) and the other point which will be y-intercept and that point y-intercept will have will be (0,-1) so now we have two points we will use the point slope form and find the slope which will be:

& \Rightarrow 0-\left( -1 \right)=m\left( -4-0 \right) \\\ & \Rightarrow 1=-4m \\\ & \Rightarrow m=-\dfrac{1}{4} \\\ \end{aligned}$$ Now that we have gotten we will use it in the slope intercept form which will give the equation of line i.e. $$\Rightarrow y=-\dfrac{1}{4}x+b$$ Now to take out the constant we will use the y-intercept point in the above equation and we will get $$\begin{aligned} & \Rightarrow -1=-\dfrac{1}{4}\left( 0 \right)+b \\\ & \Rightarrow b=-1 \\\ \end{aligned}$$ So with this our final equation will come out to be: $$\begin{aligned} & \Rightarrow y=-\dfrac{1}{4}x-1 \\\ & \Rightarrow 4y=-x-4 \\\ & \Rightarrow x+4y+4=0 \\\ \end{aligned}$$ Now the equation of line on point slope form is : $$y+1=-\dfrac{1}{4}x$$ and slope intercept form will be $$y=-\dfrac{1}{4}x-1$$ when we substitute the y-intercept values in the equation as a point.  **Note:** In the above type of question when we have been given two points and are asked to write the equation of two different forms of line equation we need to first check that if any of the equation given can be used to find the slope if yes then use that to find the slope then use that slope in other equation as the line is same.