Question

Write the equation of the line that passes through (−3,1) and (2,−1) in slope-intercept form

Answer 1

`y=-(2)/(5)x-(1)/(5)`

Explanation:

The equation of a line is y = mx + b

Where:

• m is the slope

• b is the y-intercept

First, let's find what m is, the slope of the line.

Let's call the first point you gave, (-3,1), point #1, so the x and y numbers given will be called x1 and y1.

Also, let's call the second point you gave, (2,-1), point #2, so the x and y numbers here will be called x2 and y2.

Now, just plug the numbers into the formula for m above, like this:

`m = -(2)/(5)`

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:

`y=-(2)/(5)x + b`

Now, what about b, the y-intercept?

To find b, think about what your (x,y) points mean:

• (-3,1). When x of the line is -3, y of the line must be 1.
• (2,-1). When x of the line is 2, y of the line must be -1.

Now, look at our line's equation so far:
`y=-(2)/(5)x + b`.
`b` is what we want, the -
`-(2)/(5)` is already set and x and y are just two 'free variables' sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (-3,1) and (2,-1).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for b for the particular line that passes through the two points you gave!

You can use either (x,y) point you want. The answer will be the same:

• (-3,1). y = mx + b or
  `1=-(2)/(5) * -3 + b`, or solving for b:
  `b = 1-(-(2)/(5))(-3)`.
  `b = -(1)/(5)`.
• (2,-1). y = mx + b or
  `-1=-(2)/(5) * 2 + b`, or solving for b:
  `b = 1-(-(2)/(5))(2)`.
  `b = -(1)/(5)`.

See! In both cases, we got the same value for b. And this completes our problem.

The equation of the line that passes through the points (-3,1) and (2,-1) is
`y=-(2)/(5)x-(1)/(5)`