Just in case we need to help with more party
planning for the king, let's get ourselves
some practice with solving systems
of equations with simple elimination.
And so here's the exercise right over here.
We have x minus 4y is equal to negative 18,
and negative x plus 3y is equal to 11.
So let's try to work this out.
I'll get my little scratch pad out.
This is the exact same problem.
We just rewrite it again.
It never hurts to rewrite it, get more familiarity
with the problem.
So x minus 4y is equal to negative 18.
And then we have negative x plus 3y is equal to 11.
So the thing that jumps out at my brain--
we want to eliminate one of the variables.
And we have an x here, and we have
a negative x right over there.
So that feels like those would cancel each other out
very well, so let's do that.
Let's add negative x plus 3y to the left-hand side
of x minus 4y.
And then we can add 11 to the right-hand side
because, as the second statement shows, this is equal to this.
So if we add this to the left-hand side,
we could add the same thing to the right-hand side.
But instead, we're just going to add the 11, because it's
the exact same thing as negative x plus 3y.
So we're going to add 11 to the right-hand side.
And so let's see what happens when we do that.
Well, the whole purpose is that you have an x minus an x.
Those cancel out.
And then you're just left with negative 4y plus 3y.
Negative 4 of something plus 3 of something
is you're going to have negative 1 of that something,
or you could just say negative y.
And then that's going to be equal to-- and remember,
we can still say equal because we're essentially
adding the same thing to the right-hand side
that we're adding to the left-hand side.
11 is the same thing as negative x plus 3y.
11 plus negative 18 is negative 7.
And now we're in the home stretch.
If we want to solve for y, we can divide both sides
by negative 1, or multiply both sides by negative 1,
depending on how you want to view it.
So let's divide both sides by a negative 1.
We are left with y is equal to positive 7.
But we aren't done yet.
We need to figure out what x is equal to.
So we can take this value for y, and we
should be able to substitute it back
into either of these equations to solve for x.
So let's do that.
Let's use the top equation.
So we have x minus 4 times y, but now we know that y is 7.
We know that y is 7 at the point that
satisfies both of these equations.
Why did I write y again?
4 times 7 is equal to negative 18.
Let's simplify.
You get x minus 28 is equal to negative 18.
Now to solve for x-- and we've done this many, many times
before-- we just add 28 to both sides.
So let's add 28 to the left, 28 to the right.
And we are left with x is equal to negative 18
plus 28 is the same thing as 28 minus 18, which is 10.
So we get x is equal to 10 and y is equal to 7.
And what's really good about this is you
can verify them for yourselves.
If you take this first equation, 10 minus 4 times 7, so 10
minus 28 does indeed equal negative 18.
So it checks with the first equation.
Now let's check the second equation.
Negative x, so negative 10 plus 3 times 7.
What is that equal to?
That's negative 10 plus 21, which does indeed equal 11.
So it satisfies both of them.
So now let's check our answer on the little exercise engine.
So we have y is equal to 7, x is equal to 10.
So we have x is equal to 10 and y is equal to 7.
Check our answer, and we've got it right.
We're ready to do some more cupcake planning with the king.