systems of equations

systems of equations

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.

105
00:05:20,2 --> 00:05:23,02
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.

140
00:07:07,456 --> 00:07:10,71
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.