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We're given a system of equations here, and we're told

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to solve for x and y.

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Now, the easiest thing to do here, since in both equations

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they're explicitly solved for y, is say, well, if y is equal

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to that, and y also has to equal this second equation,

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then why don't we just set them equal to each other?

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Or another way to think about it is, if y is equal to this

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whole thing right over here-- that's what that first

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equation is telling us-- and if we have to find an x and a

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y that satisfy both of these equations, if y is equal to

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that, why can't I just substitute that

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right here for y?

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And if we do that, the left-hand side of this bottom

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equation becomes negative 1/4x plus 100.

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And then that is going to be equal to this right-hand

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side-- and I'll do it in the same color-- is equal to

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negative 1/4x plus 120.

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Now, the first thing we might want to do is maybe get all of

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our x terms onto the left- or the

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right-hand side of the equation.

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And if we wanted to get rid of these x terms from the

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right-hand side, get them on the left-hand side, the best

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thing to do is to add 1/4x to both sides of this equation.

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So let me do that.

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So we're going to add 1/4x here, add 1/4x here, and you

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might already be sensing that something shady is going on.

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So let's do it.

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So negative 1/4x plus 1/4x.

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They cancel out.

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You get 0x.

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So the left side of the equation is just 100.

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And then the right side of the equation, same thing.

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Negative 1/4x plus 1/4x.

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They cancel out.

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No x's.

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And you're just left with is equal to 120.

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Which we know is definitely not the case.

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100 is not equal to 120.

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We got this nonsensical equation here,

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that 100 equals 120.

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So this type of system has no solution.

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You know it has no solution because in order for it to

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have any solution, these two numbers would have to be equal

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to each other, and they are not equal to each other.

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And if you look at the original equations, it might

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jump out at you why they have no solutions.

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Both of these lines, or both of these equations, if you

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view them as lines, have the exact same slope.

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But they have different y-intercepts.

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So if I just were to do a really quick graph here.

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That's my y-axis, that is my x-axis, so it's y and x.

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This first graph over here, its y-intercept is 100.

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Let me do it a little bit lower.

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Its y-intercept-- let's say that that is 100, so it

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intersects right there.

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And there's a slope of negative 1/4.

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So maybe it looks something like this.

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That's that first line.

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This second line-- I'll do it in pink right here-- y is

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equal to negative 1/4x plus 120, its y-intercept might be

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right here at 120.

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But it has the same slope, negative 1/4, so its slope,

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the line would look something like this.

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So you see that there are no x and y points that satisfy both

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of these equations.

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Another way to think about it.

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If y-- you take an x.

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This first equation says, OK, you take your x, multiply it

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by negative 1/4, and add 100, and that's

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going to give you y.

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Now, here we say, well, you take that same x, and you

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multiply it by negative 1/4 and add 120, and that has to

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be equal to y.

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Well, the only way that that would ever be true is if 100

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and 120 were the same number, and they're

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not the same number.

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So you're never going to have a solution of this system.

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These two lines are never going to intersect, and that's

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because they have the exact same slope.

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