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Let's say I have the equation y is equal to x plus 3.

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And I want to graph all of the sets, all of the coordinates x

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comma y that satisfy this equation right there.

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And we've done this many times before.

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So we draw our axis, our axes.

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That's my y-axis.

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This is my x-axis.

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And this is already in mx plus b form, or

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slope-intercept form.

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The y-intercept here is y is equal to 3, and the

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slope here is 1.

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So this line is going to look like this.

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We intersect at 0 comma 3-- 1, 2, 3.

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At 0 comma 3.

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And we have a slope of 1, so every 1 we go to the

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right, we go up 1.

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So the line will look something like that.

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It's a good enough approximation.

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So the line will look like this.

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And remember, when I'm drawing a line, every point on this

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line is a solution to this equation.

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Or it represents a pair of x and y that

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satisfy this equation.

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So maybe when you take x is equal to 5, you go to the

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line, and you're going to see, gee, when x is equal to 5 on

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that line, y is equal to 8 is a solution.

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And it's going to sit on the line.

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So this represents the solution set to this equation,

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all of the coordinates that satisfy y is

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equal to x plus 3.

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Now let's say we have another equation.

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Let's say we have an equation y is equal to

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negative x plus 3.

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And we want to graph all of the x and y pairs that satisfy

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this equation.

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Well, we can do the same thing.

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This has a y-intercept also at 3, right there.

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But its slope is negative 1.

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So it's going to look something like this.

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Every time you move to the right 1, you're going

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to move down 1.

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Or if you move to the right a bunch, you're going to move

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down that same bunch.

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So that's what this equation will look like.

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Every point on this line represents a x and y pair that

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will satisfy this equation.

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Now, what if I were to ask you, is there an x and y pair

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that satisfies both of these equations?

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Is there a point or coordinate that satisfies both equations?

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Well, think about it.

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Everything that satisfies this first equation is on this

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green line right here, and everything that satisfies this

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purple equation is on the purple line right there.

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So what satisfies both?

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Well, if there's a point that's on both lines, or

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essentially, a point of intersection of the lines.

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So in this situation, this point is on both lines.

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And that's actually the y-intercept.

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So the point 0, 3 is on both of these lines.

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So that coordinate pair, or that x, y pair, must satisfy

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both equations.

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And you can try it out.

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When x is 0 here, 0 plus 3 is equal to 3.

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When x is 0 here, 0 plus 3 is equal to 3.

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It satisfies both of these equations.

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So what we just did, in a graphical way, is solve a

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system of equations.

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Let me write that down.

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And all that means is we have several equations.

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Each of them constrain our x's and y's.

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So in this case, the first one is y is equal to x plus 3, and

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then the second one is y is equal to negative x plus 3.

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This constrained it to a line in the xy plane, this

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constrained our solution set to another

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line in the xy plane.

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And if we want to know the x's and y's that satisfy both of

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these, it's going to be the intersection of those lines.

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So one way to solve these systems of equations is to

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graph both lines, both equations, and then look at

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their intersection.

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And that will be the solution to both of these equations.

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In the next few videos, we're going to see other ways to

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solve it, that are maybe more

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mathematical and less graphical.

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But I really want you to understand the graphical

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nature of solving systems of equations.

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Let's do another one.

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Let's say we have y is equal to 3x minus 6.

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That's one of our equations.

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And let's say the other equation is y is equal to

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negative x plus 6.

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And just like the last video, let's graph both of these.

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I'll try to do it as precisely as I can.

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There you go.

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Let me draw some.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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And then 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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I should have just copied and pasted some graph paper here,

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but I think this'll do the job.

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So let's graph this purple equation here.

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Y-intercept is negative 6, so we have-- let me do another

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[? slash-- ?]

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1, 2, 3, 4, 5, 6.

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So that's y is equal to negative 6.

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And then the slope is 3.

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So every time you move 1, you go up 3.

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You moved to the right 1, your run is 1, your

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rise is 1, 2, 3.

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That's 3, right?

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1, 2, 3.

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So the equation, the line will look like this.

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And it looks like I intersect at the point 2

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comma 0, which is right.

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3 times 2 is 6, minus 6 is 0.

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So our line will look something

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like that right there.

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That's that line there.

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What about this line?

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Our y-intercept is plus 6.

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1, 2, 3, 4, 5, 6.

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And our slope is negative 1.

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So every time we go 1 to the right, we go down 1.

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And so this will intersect at-- well, when y is equal to

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0, x is equal to 6.

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1, 2, 3, 4, 5, 6.

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So right over there.

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So this line will look like that.

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The graph, I want to get it as exact as possible.

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And so we're going to ask ourselves the same question.

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What is an x, y pair that satisfies

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both of these equations?

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Well, you look at it here, it's going to be this point.

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This point lies on both lines.

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And let's see if we can figure out what that point is.

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Just eyeballing the graph here, it looks like we're at

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1, 2, 3 comma 1, 2, 3.

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It looks like this is the same point right there, that this

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is the point 3 comma 3.

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I'm doing it just on inspecting my hand-drawn

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graphs, so maybe it's not the exact--

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let's check this answer.

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Let's see if x is equal to 3, y equals 3 definitely

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satisfies both these equations.

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So if we check it into the first equation, you get 3 is

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equal to 3 times 3, minus 6.

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This is 9 minus 6, which is indeed 3.

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So 3 comma 3 satisfies the top equation.

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And let's see if it satisfies the bottom equation.

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You get 3 is equal to negative 3 plus 6, and negative 3 plus

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6 is indeed 3.

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So even with our hand-drawn graph, we were able to inspect

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it and see that, yes, we were able to come up with the point

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3 comma 3, and that does satisfy

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both of these equations.

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So we were able to solve this system of equations.

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When we say system of equations, we just mean many

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equations that have many unknowns.

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They don't have to be, but they tend to have more than

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one unknown.

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And you use each equation as a constraint on your variables,

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and you try to find the intersection of the equations

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to find a solution to all of them.

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In the next few videos, we'll see more algebraic ways of

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solving these than drawing their two graphs and trying to

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find their intersection points.

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