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Hi everyone!
Welcome back to integralCALC.com.
We're going to be talking about the how to find the equation of the line today.
And in this particular problem, we're given two pieces of information
in this case, two points:
one is the point (2, 3)
and the other is the origin, which is of course the point (0, 0).
So using these two points
and the slope intercept equation of the line
and slope intercept form,
we will find the
equation 1 connects these two points.
So remember that this is slope intercept form, y = mx + b,
and that m is the slope
which we find by subtracting y1 from y2
and dividing that by (x2 – x1),
remember that b is the y intercept of our line.
So, the first thing that we'll do is to go ahead and find
m, the slope,
and the way that we do that.
Remember, we have two points
and one of them is the point (x1, y1),
these two points here,
and the other is the point (x2, y2),
these two numbers right here,
so, I usually find it easier to assign
the point (x2, y2)
to whichever of our points has larger values.
In this case, our first point is (2, 3)
and the second point is (0, 0),
so, (2, 3) has larger values than it, in (0,0),
so, I will assign that to be (x2, y2)
because we're going to be subtracting
the smaller point from the larger point
and if we do that we'll end up with positive values,
and that's just easier to work with.
But it doesn’t really matter;
you could assign (x2, y2) to either
and x1 and y1 to either the points,
it's really inconsequential. But I’m going to say
that (2, 3) is (x2, y2)
and that (0, 0) is (x1, y1).
So, in that case, realize that we have the y coordinates here
and the numerator of our slope fraction.
So we'll take y2,
which in our case is 3,
remember we're treating this first point as (x2, y2),
the y coordinate there is 3,
and the y coordinate in (x1, y1) is 0,
so, we end up the numerator of our slope fraction,
we end up with (3 – 0).
For (x2 - x1), of course x2,
we grab from our first point, and that's 2,
and then subtract from that x1 which we get from the origin,
so the point is 0, so, in the denominator we end up with (2 – 0).
And we'll just plug that whole thing in from ignoring everything else for now.
The easiest way to go about this is find m first
then plug in one of the points to solve for b,
so we'll start moving in that direction now.
So first of all we can simplify our slope fraction:
(3 – 0) in the numerator gives us 3,
(2 – 0) in the denominator gives us 2,
So our slope is 3/2,
which means that every time
every time our...
the line moves up three units, right?,
the y coordinates in the numerator, it moves up 3 units,
it also moves to the right 2 units;
so the change, it's changing y, we're changing x,
the change in y is 3, it moves up 3, the change in x is 2, it moves over 2,
positive 3 and positive 2.
But anyway, that's our slope, and then we can plug in either one of these points, it doesn’t matter,
but I’ll go ahead and plug in (2, 3).
So I’ll plug in 3 for y and 2 for x,
3 for y right here, and 2 for x here.
And now solving this equation for b
will give us the last piece of information we need to write the equation of the line.
So when we multiply this out,
the 2's here will cancel, one in the denominator and one in the numerator,
and we'll be left with 3 = 3 + b,
subtracting 3 from both sides gives us b = 0.
So now if we'll look back here the equation of the line y = mx + b,
we just found that b was equal to 0 which make sense
because b is the y intercept,
and if our line intercepts the y axis at 0,
that means it passes through the origin
which was indicated in the original problem.
So, b is equal to 0 and we already found the slope was equal to 3/2;
m was equal to 3/2,
so, the equation of this line is y = (3/2) x,
and we leave y and x as they are and that's our final answer.
You can leave it in this form,
you could simplify by multiplying both sides by 2
and we get 2y = 3x or move the variables onto one side,
doesn’t really matter; unless you're ask specifically
for a slope intercept form or point slope form or whatever.
But we'll leave our equation as y equals... and then an equation in terms of x.
So, anyway, I hope that video helped you guys
and I’ll see you in the next one.
Bye!!!