find the equation for the line through the points (7,2) and (-2,-4)

Algebra
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the line through (7,2) and (-2,-4)

Jun 9th, 2015

Thank you for the opportunity to help you with your question!

You want to find the equation for a line that passes through the two points:(7,2) and (-2,-4).

First of all, remember what the equation of a line is:

y = mx+bWhere:
  • m is the slope, and
  • b is the y-intercept

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

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.

For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (7,2), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=7 and y1=2.

Also, let's call the second point you gave, (-2,-4), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=-2 and y2=-4.

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

m=
-4 - 2
-2 - 7

or...

m=
-6
-9

or...

m=2/3

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=2/3x+b
Now, what about b, the y-intercept?To find b, think about what your (x,y) points mean:
  • (7,2). When x of the line is 7, y of the line must be 2.
  • (-2,-4). When x of the line is -2, y of the line must be -4.
Because you said the line passes through each one of these two points, right?

Now, look at our line's equation so far: y=2/3x+b. b is what we want, the 2/3 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 (7,2) and (-2,-4).

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:

  • (7,2). y=mx+b or 2=2/3 × 7+b, or solving for b: b=2-(2/3)(7). b=-8/3.
  • (-2,-4). y=mx+b or -4=2/3 × -2+b, or solving for b: b=-4-(2/3)(-2). b=-8/3.
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

(7,2) and (-2,-4)

is

y=2/3x-8/3


Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 9th, 2015

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