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# Systems of equations with substitution: y=-5x+8 & 10x+2y=-2

Video transcript

Use substitution to
solve for x and y. And they give us a
system of equations here. y is equal to
negative 5x plus 8 and 10x plus 2y is
equal to negative 2. So they've set it up
for us pretty well. They already have y
explicitly solved for up here. So they tell us,
this first constraint tells us that y must be
equal to negative 5x plus 8. So when we go to the
second constraint here, every time we see a y, we say,
well, the first constraint tells us that y must be
equal to negative 5x plus 8. So everywhere we see a
y, we can substitute it with negative 5x plus 8. Because that's what the
first constraint tells us. y is equal to that. I don't want to be
repetitive, but I really want you to internalize that's
all it's saying. y is that. So every time we see a y
in the second constraint, we can substitute it with that. So let's do it. So the second equation
over here is 10x plus 2. And instead of
writing a y there, and I've said it
multiple times already, we can write a
negative 5x plus 8. The first constraint
tells us that's what y is. So negative 5x plus 8
is equal to negative 2. Now, we have one equation
with one unknown. We can just solve for x. We have 10x plus. So we can multiply it. We can distribute this 2
onto both of these terms. So we have 2 times negative
5x is negative 10x. And then 2 times 8 is 16. So plus 16 is equal
to negative 2. Now we have 10x minus 10x. Those guys cancel out. 10x minus 10x is equal to 0. So these guys cancel out. And we're just left with
16 equals negative 2, which is crazy. We know that 16 does
not equal negative 2. This is an inconsistent result. And that's because these two
lines actually don't intersect. And we could see that by
actually graphing these lines. Whenever you get something
like some number equalling some other number that
they're clearly not equal to, that means it's an
inconsistent result, It's an inconsistent system,
and that these lines actually don't intersect. So let me just graph these
just to make it clear. This first equation is already
in slope y-intercept form. So it looks something like this. That's our x-axis. This is our y-axis. And it's negative 5x plus 8,
so 1, 2, 3, 4, 5, 6, 7, 8. And then it has a very
steep downward slope. Every time you move forward
1, you have to go down 5. So it looks something like that. That's this first
equation right over there. The second equation,
let me rewrite it in slope y-intercept form. So it's 10x plus 2y is
equal to negative 2. Let's subtract 10x
from both sides. You get 2y is equal to
negative 10x minus 2. Let's divide both sides by 2. You get y is equal to negative
5x, negative 5x minus 1. So it's y-intercept
is negative 1. It's right over there. And it has the same
slope as this first line. So it looks like this. It's parallel. It's just shifted down a bit. So it just looks like that. So they're parallel lines. They have the same slope,
different y-intercepts. We get an inconsistent result. They don't intersect. And the telltale
sign of that, when you're doing it
algebraically, is you get something
wacky like this. This is why it's
called inconsistent. It's not consistent for 16
to be equal to negative 2. These don't intersect. There's no solution to both of
these constraints, no x and y that satisfies both of them.