How to write an equation for the line in point-slope form and then re-write in slope-intercept form.

Question

A line passes through the given points (-1,0),(1,2). Write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form.

Answer 1

The first thing we want to do is to find the slope between these two points.  The equation for the slope, m, is:

m=rise/run=(y₂-y₁)/(x₂-x₁).

Let the first point (-1,0)=(x₁,y₁) and the second point (1,2)=(x₂,y₂), then

m=(2-0)/(1-(-1))=2/2=1, so the slope between these two points is 1.

Now, we can write the equation of the line in point-slope form, where m is the slope:

y-y₁=m(x-x₁)

y-0=1(x-(-1))

y-0=1(x+1) or y=1(x+1), which is the equation of the line in point-slope form.  Note: Instead of (x₁,y₁), you could also use (x₂,y₂).  Your point-slope form equation will then look like: y-2=1(x-1), which will still give you the same equation for the slope-intercept form once you rearrange the equation.

The equation for slope-intercept form is, where m is the slope and b is the y-intercept:

y=mx+b

So we just want to rearrange our equation of the line in point-slope form so that it looks like the equation of the line in slope-intercept form.  To do that, we re-write y=1(x+1) as:

y=1x+1 where the 1 was distributed through the parenthesis.  So the equation of the line in slope-intercept form is:

y=x+1

And for this equation, we see that b=1, so the equation goes through the y-axis at y=1.

Hope that helps!