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# 3-variable linear system word problem

Sal solves a word problem about the angles of a given triangle by modeling the given information as a system of three equations and variables. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Solve the following application
problem using three equations with three unknowns. And they tell us the
second angle of a triangle is 50 degrees less than
four times the first angle. The third angle is 40
degrees less than the first. Find the measures
of the three angles. So let's draw ourselves
a triangle here. And let's call the first angle
a, the second angle b, and then the third angle c. And before we even look
at these constraints, one property we
know of triangles is that the sum of their
angles must be 180 degrees. So we know that a plus b plus
c must be equal to 180 degrees. Now, with that out of
the way, let's look at these other constraints. So they tell us the second
angle of a triangle-- let me do that in
another color-- they tell us the second
angle of a triangle is 50 degrees less than
four times the first angle. So we're saying b
is the second angle. So they're saying, the
second angle of a triangle is 50 degrees less than
four times the first angle. So 4 times the first
angle would be 4a, we're calling a the first angle. So 4 times the
first angle is 4a. But it's 50 degrees less
than that, so minus 50. Now the next constraint they
give us, the third angle is 40 degrees less
than the first. So the third angle is 40
degrees less than the first. So the first angle, a, is
going to be 40 degrees less than that. So we have three equations
with three unknowns. And so we just have
to solve for it. And let's see, what's a
good first variable to try to eliminate? And just to try to visualize
that a little bit better, I'm going to bring these a's
onto the left hand side of each of these equations over here. So I'm going to rewrite
the first equation. We have a plus b plus
c is equal to 180. And then this equation,
if we subtract 4a from both sides of this
equation, if we subtract 4a we have negative 4a plus
b is equal to negative 50. And then this equation
right over here, if we subtract a
from both sides we get negative a plus c
is equal to negative 40. I just subtract a
from both sides. So we now want to
eliminate variables. And we already have--
this third equation here is only in
terms of a and c. This is only in
terms of a and b. And this first one is
in terms of a, b, and c. So if we could-- let's see, this
is already in terms of a and c. If we could turn these
first two equations, if we can use the information
in these first two equations to end up with an
equation that's only in terms of
a and c, then we could use whatever
we end up with along with this third equation
right over here, and we'll have a system of two
equations with two unknowns. So let's do that. So if we wanted to just end up
with an equation only in terms of a and c using
these first two, we would want to
eliminate the b's. So we could multiply
one of these equations times negative 1, and one
of these positive b's will turn into a negative b. So let's do that. Let's multiply this first
equation over here, let's multiply it times negative 1. So it'll become
negative a minus b minus c is equal
to negative 180. And then we have this
green equation right over here, which is really just
this equation, just rearranged. So we have negative 4a plus
b is equal to negative 50. And now we can add
these two equations. Actually, let me do
that in the other color, just so you see where
that's coming from. I'll do it in that green color. So this is negative 4a plus
b is equal to negative 50. We can add these two up now. And we get negative a
minus 4a is negative 5a. The b's cancel out. We have a minus c is equal
to negative 180 minus 50 is negative 230. So now using these
top two equations we have an equation only
in terms of a and c. We have another equation
only in terms of a and c. And it looks like if we
add them together their c's will cancel out. So let me just rewrite
this equation over here. And you have to be
careful that you're using all of the equations. Otherwise you'll do
a circular argument. You have to be careful that over
here this first equation came from these two over here. Now I want to combine that
with this third constraint, s constraint that's not already
baked into this equation right over here. So we have negative a plus
c is equal to negative 40. And we add these two equations. Negative 5a minus
a is negative 6a. The c's cancel out. And then you have
negative 230 minus 40. This is equal to negative 270. We can divide both
sides by negative 6. And we get a is equal
to negative 270 over 6. Let me see how many times--
let me see something. 270 is divisible
by both 3 and 2. So it should be divisible by 6. So let me just divide it. The negative signs
obviously will-- a negative divided by a negative is
going to be a positive. And if we take 6 into 270,
6 goes into 27 4 times. 4 times 6 is 24. We subtract. We get 3, bring down the 0. 6 goes into 30 five times. So we get a is equal to 45. Now let's look at
the other ones. We can substitute back
in to solve for c. c is equal to a
minus 40 degrees. So that is equal to-- let
me write it right over here in yellow-- so c
is equal to 45 minus 40, which is equal to 5 degrees. So, so far we have a
is equal to 45 degrees, c is equal to 5 degrees. And then you could
substitute into either one of these other ones
to figure out b. We could use this one
right over here in green. B is equal to 4a minus 50. So b is going to be equal to
4 times 45 is-- let's see, 2 times 45 is 90. So 4 times 45 is 180. So it's going to
be 180 minus 50, by this equation
right over here, which is equal to 130 degrees. So we get b is equal
to 130 degrees. And then we can-- let
me write over here. So a is equal to 45. So if I wanted to
draw this triangle it would actually look
something like this. a is a 45 degree angle, b is a
130 degree angle, and c is 5. So it'll look
something like this. It will look
something like this, where this is a at 45
degrees, b is 135 degrees, and then c is 5 degrees. And you can verify
that it works. One, you can just
add up the angles. 45 plus 5 is 50. Oh, sorry, this isn't 135. It's 130. We solved it right over here. It's 130 and this is 5. So when you add them
all up-- 45 plus 130 plus 5-- that does
indeed equal 180 degrees. 45 plus 5 is 50 plus 130. So this does
definitely equal 180. So it meets our
first constraint, Then on our second
constraint, b needs to be equal to 4a minus 50. Well, 4 times a is 180
minus 50 is 130 degrees. So it meets our
second constraint. And then our third constraint,
c is a minus 40 degrees. Well, a is 45, c is 5. So if you subtract 40 from
45 you get 5, which is c. So it meets all of our
constraints and we are done.