Welcome to the presentation on systems of linear equations.
So let's get started and see what it's all about.
So let's say I had two equations now.
The first equation let me write it as 9x minus
4y equals minus 78.
And the second equation I will write as 4x plus
y is equal to mine 18.
Now what we're going to do now is we're actually going to
use both equations to solve for x and y.
We already know that if you have one equation, it has one
variable, it is very easy to solve for that one variable.
But now we have to equations.
You can almost view them as two constraints.
And we're going to solve for both variables.
And you might be a little confused.
How does that work?
Is it just magic that two equations can solves
for two variables?
Well it's not.
Because you can actually rearranged each of these
equations so that they look kind of in normal y
equals mx plus b format.
And I'm not going to draw these actual two equations because I
don't know what they look like, but if this was a coordinate
axis-- and I don't know what that first line actually does
look like, we could do another model where we figured it out
--but lets just say for sake of argument, that first line all
the x's and y's that satisfy 9x minus 4y equals negative
78, let's say it looks something like that.
And let's say all of the x's and y's that satisfy that
second equation, 4x plus y equals negative 18, let's say
that looks something like this.
Right?
So, on the line is all of the x's and y's that satisfy this
equation, and on the green line are all the x's and y's
that satisfy this equation.
But there's only one pair of x and y's that satisfy both
equations, and you can guess where that is, that's
right here right.
Whatever that point is-- I'll do it in pink for emphasis.
Whatever this point is, notice it's on both lines.
So whatever x and y that is would be the solution to
this system of equations.
So let's actually figure out how to do that.
So what we want to do is eliminate a variable, because
if you can eliminate a variable then we can just solve for
the one that's left over.
And the way to do that-- let's see, I want to eliminate, I
feel like eliminating this y, and I think you'll get
an intuition for how we can do that later on.
And the way I'm going to do that is I'm going to make
it so that when I had this to this, they cancel out.
Well, they don't cancel out right now, so I have to
multiply this bottom equation by 4, and I think it'll be
obvious why I'm doing it.
So let's multiply this bottom equation by 4.
And I get 16x plus 4y is equal to 40 plus 32 minus 72.
Right?
All I did is I multiplied both sides of the
equation by 4, right?
And you have to multiply every term because
it's the distributive property on both sides.
Whatever you do to one side you have to do to the other.
Let me rewrite top equation again.
And I'll write in the same color so we can keep
track of things.
9x minus 4y is equal to minus 78.
OK, well now, if we were to add these two equations, when you
add equations, you just add the left side and you
add the right side.
Well when you add, you have 16x plus 9x.
Well that equals 25x.
Right?
16 plus 9.
4y minus 4, that just equals 0.
So that's plus 0 equals, and then we have minus 72 minus 78.
So, let's see that's minus 150, minus 150, right?
Just adding them all together.
So we have 25x equals 150.
Well, we could just divide both sides by 25 or multiply both
sides by 1/25, it's the same thing.
And you get x equals-- that's a negative 150
--x equals minus 6.
There we solved the x-coordinate.
Now to solve the y-coordinate we can just use either one of
these equations up at top.
So let's use this one, it seems a little bit,
marginally simpler.
So we just substitute the x back in there and we get
4 time minus 6 plus y is equal to minus 18.
Go up here.
4 times minus 6 we get minus 24 plus y is equal to minus 18.
And then get y is equal to 24 minus 18.
So y is equal to 6.
So these two lines or these two equations, you could even say,
intersect at the point x is m inus six and y is plus 6.
So they actually intersect someplace around here instead.
I drew these, the line probably look something more like that.
But that's pretty cool, no?
We actually solved for two variables using two equations.
Let's see how much time I have.
I think we have enough time to do another problem.
So let's say I had the points-- and I'm going to write them in
two different colors again --minus 7x minus 4y equals 9,
and then the second equation is going to be x plus
2y is equal to 3.
Now if I were doing this as fast as possible, I'd probably
multiply this equation times 7 and it would automatically
cancel out.
But that's easy way.
I'm going to show you that sometimes you might have to
multiply both equations-- actually, not in this case.
Actually let's just do it the fast way real fast.
So let's multiply this bottom equation by 7.
And the whole reason why I want to the, multiply it with 7,
because I want this to cancel out with this.
If you multiply it by 7 you get 7x plus 14y is equal to 21.
Let's write that first equation down again.
Minus 7x minus 4y is equal to 9.
Now we just add.
This is a positive 7x, it just always looks like a negative.
OK, so that's 0.
14 minus 4y plus 10y is equal to 30.
y is equal to 3.
Now we just substitute back into either equation,
lets do that one.
x plus 2 times y, 2 times 3.
x plus 6 equals 3.
We get x equals negative 3.
That one was super easy.
The intercept.
Hope I didn't do it to fast.
Well, you can pause it and watch it again if you have.
OK, so these two lines intersect at the point
negative 3 comma 3.
Let's do one more.
Hope this one's harder.
I think it will.
OK, negative 3x minus 9y is equal to 66.
We have minus 7x plus 4y is equal to minus 71.
So here it's not obvious.
What we have to do is, let's say we want to cancel
out the y's first.
What we do is we try to make both of them equal to the least
common multiple of 9 and 4.
So, if we multiply the top equation by 4 we get--
I'll do it right here.
Let's multiply it by 4.
Times 4.
We'll get minus 12x minus 36y is equal to 4 times
240 plus 24 is 264.
Right, I hope that's right.
We multiply the second equation by 9.
So it's minus 63x plus 36y is equal to, let's see, 639.
Big numbers.
639.
OK, now we add the two equations.
Minus 12 minus 63 thats minus 75x-- these cancel out --equals
264, let's see what's 639 minus 264.
See I do this in real time.
I don't use some kind of solution manual or something.
13 and 5, 70.
I don't know if I'm right, but we'll see.
Since it's actually the negative 639, this is minus
375, and I know that seventy five goes into 300 4
times, so x is equal to 5.
75 times 5 is 375.
We just divided both sides by 75.
So if x is 5 we just substitute it back into-- let's
use this equation.
So we get minus 3 times 5 minus 9y is equal to 66.
We get minus 15 minus 9y equals 66.
Minus 9y is equal to 81.
And then we get y is equal to minus 9.
So the answer is 5 comma minus 9.
I think you're ready to do some systems of equations now.
Have Fun.