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# Systems of equations with substitution: y=4x-17.5 & y+2x=6.5

Video transcript

We have this system of
equations, y is equal to 4x minus 17.5, and y plus
2x is equal to 6.5. And we have to solve
for x and y. So we're looking for x's
and y's that satisfy both of these equations. Now, the easiest way to think
about it is we've already solved for y in this
top equation. Let me write it again. I'll write it in pink. We have y is equal
to 4x minus 17.5. So this first equation is
telling us, literally, by this constraint, y should be
4 times x minus 17.5. Now, the second equation says
whatever y is, we had 2 times x, and that should be 6.5. Well, the y here also has to
meet this constraint up here. It also has to meet the
constraint that it has to be 4 times x minus 17.5. So what we can do is, is we can
substitute this value for y into this equation. Let me be clear what
I'm doing. The second equation here is
y plus 2x is equal to 6.5. We know that y has to be equal
to this thing right here. y has to be equal to
4x minus 17.5. So let's take 4x minus 17.5,
and substitute y with that. So let's put that right there. So if we were to do that, if we
were to replace this y with 4x minus 17.5, because that's
what the first equation is telling us, then we get
4x minus 17.5, plus 2x is equal to 6.5. And now we have a single linear equation with one unknown. Let's solve for x. So first we have our x
terms. We have a 4x, and we have a 2x. We can group them or
add them together. 4x plus 2x is 6x. And then we have 6x minus
17.5 is equal to 6.5. Then we can get the 17.5 out
of the way by adding it to both sides of the equation. So this is negative 17.5, so
let's add positive 17.5 to both sides of this equation. And we are left with the
left-hand side is just going to be 6x, because these
guys cancel out. 6x is going to be equal to--
and 6.5-- see, 6 plus 17 is 23, and then 0.5
plus 0.5 is 1. So this is going to be 24. And then we can divide both
sides of this equation by 6. And you are left with x is equal
to 24 over 6, which is the same thing as 4. So we figured out the x value
for the x and y pair that satisfy both of these
equations. Now we need to figure
out the y value. And we can do that by taking
this x and putting it back into one of these equations. We can do it in to either one. We should get the
same y value. So let's just do this
top one up here. So if we assume x is equal to 4,
this top equation tells us y is equal to 4 times x, which
in this case is 4, minus 17.5. Well, this is equal to 16 minus
17.5, which is equal to negative 1.5. So y is equal to negative 1.5. So the solution to this system
is x is equal to 4, y is equal to negative 1.5. And you can even verify that
these two, they definitely work for the top one if you put
4 times 4, minus 17.5, you get negative 1.5. But they also work for
the second one. And let's do that. In the second one, if you take
negative 1.5, plus 2 times x-- plus 2 times 4-- what
does that equal? That's negative 1.5 plus 8. Well, negative 1.5
plus 8 is 6.5. So this x and y satisfy both
of these equations.