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Let's do some more systems of equations problem.

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In this video, we're going to encounter systems that might

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have no solutions, or that might have an infinite many

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solutions, and we'll label them with words.

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So let's start with one.

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Let's say we have 3x minus 4y is equal to 13.

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Let's say my other equation in my system is y is equal to

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negative 3x minus y.

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So the first thing-- this is kind of in a strange form

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right here.

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I want to get into the standard form and maybe I'll

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do elimination for this systems. Let me rewrite the

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top equation.

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We have 3x minus 4y is equal to 13.

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And let me rearrange this bottom equation here.

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So if I were to add 2y-- well, let me subtract y from both

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sides of this equation.

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So if I subtract y from both sides of this equation, it

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becomes 0 is equal to negative 3x minus 2y, or negative 3x

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minus 2y is equal to 0.

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So let me write that over here.

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And it looks nice because I have a negative 3x here, I

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have a positive 3x here.

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Looks well suited for elimination.

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So, I have negative 3x minus 2y is equal to 0.

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So let's add the left-hand side of this equation to the

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left-hand side of the yellow equation.

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And we're going to add 0 to the right-hand side of the

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yellow equation and we're essentially

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adding 0 to both sides.

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We're adding the same quantity to both sides, which we can

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always do with an equation.

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So, the left hant-side, the 3x cancels out with a 3x and

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we're left with negative 4y minus 2y.

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You get negative 6y is equal to 13.

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Divide both sides by negative 6.

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We are left with y is equal to negative 13/6.

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Now let's solve for x.

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And we can solve for x using either of these equations.

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Let's use that top one, just for fun.

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So we have 3 times x minus 4 times negative

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13/6 is equal to 13.

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Now we have a negative times a negative, so those are both

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going to become positives.

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And then the 4/6, that's the same thing as 2/3.

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So this becomes 3x plus 2 times 13 which is 26/3, is

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equal to 13.

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Instead of 13, since I'm about to subtract 26/3 from both

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sides, let me rewrite 13 as 39/3.

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right?

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39 divided by 3 is 13.

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So, let me subtract 26/3 from both sides.

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The left-hand side becomes 3x-- these cancel out-- is

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equal to 39 minus 26 is 13/3.

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And then we're going to want to divide both sides by 3.

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Or you can view it as multiplying both sides by 1/3.

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The left-hand side, we're just left with an x.

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The right-hand side, x is going to be equal to 13/9.

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So this system had a well defined solution.

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The solution is x is equal to 13/9 and y is equal to

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negative 13/6.

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It only has one solution.

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So if you think of these as lines, these two lines

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intersect in exactly one point.

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And a system like this, where it has exactly one solution,

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is called a consistent system of equations.

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And everything we've been doing so far has been

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consistent systems. Let's see if we can stumble upon

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something that's maybe a little less consistent.

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Let's say we have a system.

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Let's say it's 5x minus 4y is equal to 1.

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And let's say we have negative 10x plus 8y is equal to

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negative 30.

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Once again, I'm tempted to do elimination here, because I

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have a negative 10x and I have a 5x.

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If I take this top equation and I multiply it by 2, I'll

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get 10x minus 8y is equal to 2, right?

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I just multiplied both sides by 2.

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And if we add the left-hand sides, we'll get 0x plus 0y is

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equal to negative 28.

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So we essentially get 0 is equal to negative 28.

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That's crazy.

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We know that that's not true.

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This can never be true.

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We're getting an inconsistent statement.

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We're getting a weirdo statement, and that's because

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this has no solution.

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When you solve a system of equation, it doesn't matter

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how you do it, whether it's through substitution or

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whether it's through elimination, like I did here.

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When you get one of these statements where 0 equals

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negative 28 or 5 is equal to 7 or two things that clearly

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don't equal each other, when they essentially have to equal

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each other in order for the system to work, we call that

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an inconsistent system.

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And it will have no solution.

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So what does it mean for both of these

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equations to have no solution?

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Let's actually graph these.

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I think you'll have a better feel for what it means not to

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have a solution.

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So the first equation is 5x minus 4y is equal to 1.

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Let me put it into slope-intercept form.

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So if we subtract 5x from both sides, we get negative 4y is

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equal to negative 5x plus 1.

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Now if we divide both sides by negative 4, you get y is equal

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to negative 5/4x minus 1/4, right?

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1 divided by negative 4.

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So this is the first equation, right over here, in

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slope-intercept form.

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Now let me write the second equation in

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slope-intercept form.

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We have negative 10x plus 8y is equal to negative 30.

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Let's add 10x to both sides.

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You get 8y is equal to 10x minus 30, and let's divide

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both sides by 8.

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You'll get y is equal to 10/8, is the same thing as 5/4.

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5/4x minus 30/8.

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30/8 is the same thing as 15/4.

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Oh, and actually I made a mistake here.

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When we divide both sides of this equation by negative 4,

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negative 5 divided by negative 4 is positive 5/4.

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So I shouldn't have had a negative there.

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I almost made a blunder.

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So there should be no negative there.

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Then a 1 divided by negative 4, is negative 1/4.

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So going back to the two equations, what do you notice?

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Well, when you put them in slope-intercept form, they

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both have the exact same slope, 5/4.

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but they have different y-intercepts.

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So what would their graphs look like?

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Let me graph them.

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Let's say that that is my y-axis.

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That is my x-axis.

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This equation, its y-intercept is at 0 negative 1, 4.

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Maybe that's that point right there.

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And it goes up at 5/4x.

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That's a little bit more than one.

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It's a 1.25x.

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Every time you go to the right 1, you're going up 1.25.

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So this line is going to look something like this.

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I'm just drawing it rough.

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I want you to get the general idea.

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That's what that line looks like.

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Now this line, its y-intercept is at negative 15/4.

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15/4 is what?

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3 3/4.

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So if this is negative 1/4, its y-intercept is going to be

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way down here someplace.

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I'll do it in the same color.

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Way down here someplace.

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Let me continue my x-axis down.

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This would be at negative 15/4.

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But its slope is the exact same thing.

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

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up by 5/4, so its slope is going to be

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the exact same thing.

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So what do you notice about these two lines?

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They are parallel.

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They have the same slope, different y-intercepts, so

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they will never intersect.

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These two lines will never intersect.

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Which means that there is no point on the coordinate plane

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on the x-, y-coordinate plane that satisfies

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both of these equations.

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Remember, this line represents all of the points that

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satisfied this equation.

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This line represents all of the points that

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satisfied that equation.

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Notice, no point satisfies both.

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There's no point of intersection and that's why

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this was an inconsistent system.

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Let's do one more.

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Let's say I have 4x plus 5y is equal to 0, and I have 3x is

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equal to 6y plus 4.5.

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Actually, let me do a slightly different one, because I want

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to show you all of the different types that we can

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see in systems of equations.

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Let me clear this.

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Let's say my first system is 3x minus 7y is equal to 1.

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And let's say my other equation in my system is

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negative 6x plus 14y is equal to negative 2.

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So, let's try to find the x's and y's that satisfy this

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equation here.

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And just for a change of pace, let's do some substitution.

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Although this is very tempting to do elimination here, let's

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do substitution.

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You get 3x.

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Let's solve it for x.

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Actually, let's just do elimination, because this is

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just so glaringly prepared for elimination, let's

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just do it that way.

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So let's multiply this top equation by 2.

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And what do we get?

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We get 6x minus 14y is equal to 2 right?

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I just multiplied every term on both sides by 2.

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And now let's add the left sides together.

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You get 0 plus 0.

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And on the right hand side, you get is equal to 0.

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You get 0 is equal to 0, which is always going to be true.

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This type of system is called a dependent system.

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So remember, when you get a nice clean solution, that's a

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consistent system.

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When you get something crazy like 0 equals 1, that's an

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inconsistent system.

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That means the lines are parallel.

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When you get 0 equals 0, or 1 is equal to 1 or anything like

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that, you're dealing with the dependent system.

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Which really means that these are the exact same lines, even

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though they might look a little bit different.

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And to verify that, let's put them both into

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slope-intercept form.

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So this top line, you have 3x minus 7y is equal to 1.

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Let's subtract 3x from both sides.

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You get negative 7y is equal to negative 3x plus 1.

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Now let's divide both sides by negative 7.

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You get y is equal to positive 3/7x minus 1/7.

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That's that first equation.

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Now let's put the second equation into

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slope-intercept form.

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You have negative 6x plus 14y is equal to negative 2.

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Let's add 6x to both sides of the equation.

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So you get 14y is equal to 6x minus 2.

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Then divide both sides of the equation by 14.

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You get y is equal to 6/14x minus 2/14.

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Well, this is the same thing.

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6/14 is the same thing is 3/7x minus 1/7.

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Notice, they are really the exact same equation.

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So if you want to find x's and y's that satisfy both, let's

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think about it.

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Let's graph it.

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So if that is my coordinate plane, that's the y-axis, that

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is the x-axis.

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This graph is going to look something like this.

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I'm going to draw it very roughly.

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It might look something like that, where its slope is 3/7.

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So this line is going to look something like that.

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That line looks exactly like that.

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It's the same exact line.

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So when you say, well, what are the x's and y's that

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satisfy both of these equations?

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Well, it's every x and y that's on these points.

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It's that x and y.

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That coordinate.

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That coordinate.

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That coordinate.

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There are an infinite number of solutions.

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And when we use the word dependent -- because you can

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get to one of these equations from the other.

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These equations are dependent on each other.

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You can just scale one or the other and rearrange it, and

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they equal each other.

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So here you have infinite solutions.

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Anything that satisfies one line will satisfy the other.

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So you just pick an x.

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When x is 1, you get 3/7 minus 1/7 that's 2/7.

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So 1, 2/7 satisfies both equations.

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If you pick x is equal to 0.

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0, negative 1/7 satisfies both equations.

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And you could pick an infinite number of values for x, solve

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for y, and those coordinates will satisfy both equations.

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Let me review this a little bit.

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So we started off just with the plain vanilla.

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When you actually get a solution that is consistent.

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These lines actually intersect in one point.

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Then you have the situation where you get something crazy,

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when you solve your system of equations.

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0 is equal to negative 28.

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Definitely not true.

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This is an inconsistent system.

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It has no solution, which means that

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these lines are parallel.

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They never intersect.

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And then finally, if you get something that's always true--

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that's just kind of silly how true it is-- this is a

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dependent system.

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These are going to be the same line.

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Then you can verify it by putting both of them into

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slope-intercept form.