Question

how do you find an equation of the line containing the given pair of points (-1,-5)and(-6,-7)

Answer 1

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

y = mx+b
Where:
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, (-1,-5), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=-1 and y1=-5.
Also, let's call the second point you gave, (-6,-7), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=-6 and y2=-7.

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

m=
-7 - -5
-6 - -1
or...

m=
-2
-5
or...

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=2/5x+b
Now, what about b, the y-intercept?
To find b, think about what your (x,y) points mean:
(-1,-5). When x of the line is -1, y of the line must be -5.
(-6,-7). When x of the line is -6, y of the line must be -7.
Because you said the line passes through each one of these two points, right?
Now, look at our line's equation so far: y=2/5x+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 (-1,-5) and (-6,-7).

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:

(-1,-5). y=mx+b or -5=2/5 × -1+b, or solving for b: b=-5-(2/5)(-1). b=-23/5.
(-6,-7). y=mx+b or -7=2/5 × -6+b, or solving for b: b=-7-(2/5)(-6). b=-23/5.
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
(-1,-5) and (-6,-7)

is

y=2/5x-23/5