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# Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120

Learn to sol the system of equations y = -1/4x + 100 and y = -1/4x + 120 using substitution. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're given a system of
equations here, and we're told to solve for x and y. Now, the easiest thing to do
here, since in both equations they're explicitly solved for y,
is say, well, if y is equal to that, and y also has to equal
this second equation, then why don't we just set
them equal to each other? Or another way to think about
it is, if y is equal to this whole thing right over here--
that's what that first equation is telling us-- and if
we have to find an x and a y that satisfy both of these
equations, if y is equal to that, why can't I just
substitute that right here for y? And if we do that, the left-hand
side of this bottom equation becomes negative
1/4x plus 100. And then that is going to be
equal to this right-hand side-- and I'll do it in the
same color-- is equal to negative 1/4x plus 120. Now, the first thing we might
want to do is maybe get all of our x terms onto the
left- or the right-hand side of the equation. And if we wanted to get rid
of these x terms from the right-hand side, get them on the
left-hand side, the best thing to do is to add 1/4x to
both sides of this equation. So let me do that. So we're going to add 1/4x here,
add 1/4x here, and you might already be sensing that
something shady is going on. So let's do it. So negative 1/4x plus 1/4x. They cancel out. You get 0x. So the left side of the
equation is just 100. And then the right side of
the equation, same thing. Negative 1/4x plus 1/4x. They cancel out. No x's. And you're just left with
is equal to 120. Which we know is definitely
not the case. 100 is not equal to 120. We got this nonsensical
equation here, that 100 equals 120. So this type of system
has no solution. You know it has no solution
because in order for it to have any solution, these two
numbers would have to be equal to each other, and they are
not equal to each other. And if you look at the original
equations, it might jump out at you why they
have no solutions. Both of these lines, or both
of these equations, if you view them as lines, have
the exact same slope. But they have different
y-intercepts. So if I just were to do a
really quick graph here. That's my y-axis, that is my
x-axis, so it's y and x. This first graph over here,
its y-intercept is 100. Let me do it a little
bit lower. Its y-intercept-- let's say
that that is 100, so it intersects right there. And there's a slope
of negative 1/4. So maybe it looks something
like this. That's that first line. This second line-- I'll do it
in pink right here-- y is equal to negative 1/4x plus 120,
its y-intercept might be right here at 120. But it has the same slope,
negative 1/4, so its slope, the line would look something
like this. So you see that there are no x
and y points that satisfy both of these equations. Another way to think about it. If y-- you take an x. This first equation says, OK,
you take your x, multiply it by negative 1/4, and
add 100, and that's going to give you y. Now, here we say, well, you
take that same x, and you multiply it by negative 1/4 and
add 120, and that has to be equal to y. Well, the only way that that
would ever be true is if 100 and 120 were the same
number, and they're not the same number. So you're never going to have
a solution of this system. These two lines are never going
to intersect, and that's because they have the
exact same slope.