y=1/2x+1/2
m=1/2
Step-by-step explanation:
You want to find the equation for a line that passes through the two points:
(5,3) and (-3,-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, (5,3), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=5 and y1=3.
Also, let's call the second point you gave, (-3,-1), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=-3 and y2=-1.
Now, just plug the numbers into the formula for m above, like this:
m=
-1 - 3 over
-3 - 5
or...
m=
-4 over
-8
or...
m=1/2
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=1/2x+b
Now, what about b, the y-intercept?
To find b, think about what your (x,y) points mean:
(5,3). When x of the line is 5, y of the line must be 3.
(-3,-1). When x of the line is -3, y of the line must be -1.
Because you said the line passes through each one of these two points, right?
Now, look at our line's equation so far: y=1/2x+b. b is what we want, the 1/2 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 (5,3) and (-3,-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:
(5,3). y=mx+b or 3=1/2 × 5+b, or solving for b: b=3-(1/2)(5). b=1/2.
(-3,-1). y=mx+b or -1=1/2 × -3+b, or solving for b: b=-1-(1/2)(-3). b=1/2.
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
(5,3) and (-3,-1)
is
y=1/2x+1/2
