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This is a video to accompany objective 2.5 that has to do with solving linear equations.
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We want to be able to solve them numerically and graphically, which we kind of already know
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how to do.
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Well, kind of, we do know how to do that.
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We don't know too much about how to solve them symbolically, but you're going to learn
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that in this objective.
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And symbolically actually has a name.
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They're called literal equations, literal, not linear, but literal.
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So let's see.
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Way back when, who knows, seventh grade, maybe even, you learned to solve linear equations
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very similar to this one.
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They did crazy stuff where they said you need to put a minus 8 over here and a minus
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8 over here.
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So then you had to draw a line underneath that.
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And then that gave you what 12 over here and 2 over here.
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And then you're supposed to put a divided by 2 and a divided by 2.
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And you see then you get like 6, right?
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OK, so I know you know how to do this.
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You've done solving of linear equations before when there's only one unknown.
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Now you know that unknown was basically the input of some function, right?
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That's what it is.
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You didn't know at the time, but now you know it's the input of some function, this
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case T. And in this class, you've also learned another graphical technique.
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And what's that?
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You would put the function P in here.
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This is the function P from above this guy.
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You recognize it.
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Here's the, we're not going to call it the intercept, right?
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Here's the initial value.
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And the rate is the slope of this line, right?
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Which happens to be 2.
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OK, and you learn a technique which said if you put in the output of 20 in as a second
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function, you get a horizontal line.
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And then you use the intersect command, and what the intersect command does is allows the
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calculator to tell you what the corresponding input of that is.
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That intersection point in it's 6, right?
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So we've already done this in the calculator.
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And using the calculator to solve a simple linear equation like this is maybe silly, right?
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Maybe silly.
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If you already had this in here though, if you already had this line in here in which
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you will have when you learn to do linear regression, you already have the line there, then maybe
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putting this in and finding the intersection is the fastest way, faster than going back
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to a hand calculation.
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OK, but let's do something different here when we're talking about solving linear equations.
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Let's solve a linear equation that is literal.
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So look, it's got two adjectives on it.
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It's a literal linear equation.
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That's exactly what this is.
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OK, it's an equation, this one right here, it's an equation that has what?
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That has no values for its parameters.
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It's an equation that has no values for its parameters.
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Let me zoom in here a little bit.
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OK, so here is a function with its parameters, but here is a literal,
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literal linear equation.
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It's linear in what?
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It's linear in T. It also happens to be linear in A and B and Q. It's linear in all of them,
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because they're all of the first power.
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But we know from the previous line that T is probably the most interesting one, because
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it's the input.
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But to the truth, they're all linear in every single one of them, and it's a literal
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equation because there's no numbers, no numbers.
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So what if I want to solve for T now?
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What would I do?
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Well, I kind of follow the same solving techniques that I had when I was in seventh grade up
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above, but now there's no numbers to do the actual arithmetic.
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So I have to leave everything in its algebraic form.
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I can't actually subtract B from Q again, because I don't know what the numbers are.
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So you notice something?
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I put the B over on this side.
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It's a negative B. Now that's something that you have to remember from solving.
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That when you move a term to the other side, you change its sign.
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And when you want to get rid of a coefficient, you multiply it.
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And multiply it by its multiplicative inverse.
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Don't divide by A, say multiply by 1 over A. Then you won't be confused when there's
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a fraction there, by the way.
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So let's see.
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I divide by A, quote A, or better multiply both sides by the multiplicative opposite
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of A, which is 1 over A, and the A disappears in front of T.
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Now it's over here, and now I'm writing it in the regular form, the one up above.
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This is in line, right, for your calculator.
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And because we normally write the solved for symbol on the left, I just swapped the order
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of these two guys.
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They're totally equivalent, right?
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OK, so that actually tells us how to solve simple linear equations, right?
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So now we have a little thing we can say, because the input T is the output minus the what?
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Minus the constant in the linear function.
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That's B divided by the what?
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Divided by the coefficient, the linear coefficient.
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So if you ever see one of these again, you don't have to solve it.
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You don't have to solve this.
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Like if I go back up here, if I go back up here to this one, what's the answer to this
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one?
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It's what?
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Subtract 20 for main divided by 2, right?
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That's what that just said.
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Take the output, subtract the constant divided by the coefficient.
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So if I do that, no surprise, I get 6.
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So great.
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One of the nice things about solving a literal equation is we get a formula so that we don't
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have to solve it again.
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If we just know Q, B, and A, we just plug them in.
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We don't have to solve for T every time over and over and over again.
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It's beautiful, right?
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If you've been shown that, maybe you would have never solved another linear equation again.
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Okay, now what I want to do is I want to do something a little harder.
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About a harder linear literal equation, a harder one.
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Okay.
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I'm thinking of a function, a function P. Here's the model for that function.
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Here's the rule, if you will.
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Okay.
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Now I'm dropping out the middle part.
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Now I'm making my equation.
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Now I'm going to solve for T. So I'm going to solve for T. That's my mission.
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Solve for T.
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So how do I solve for T?
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Well, the procedure is just a little different than you're used to.
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Well, first of all, both of these terms have T in them.
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So I'm going to leave them together.
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And this guy is a constant relative to these two terms right here that have T. That's apparently
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the input.
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So it doesn't have any T in this term.
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So I'm going to move that to the other side becomes Q plus C. Oh, and I missed this step
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right here, which was, I needed to distribute the minus sign on each of those terms.
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I come out, expand is what I'm doing, expand.
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Now I'm collecting the like terms, Q and C are like terms because they don't have T in
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them.
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So I'm collecting them on this side.
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Now the terms that do have T in them are collected next to each other.
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Now I'm going to do, I am going to factor.
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I'm going to find the greatest common factor, right?
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Which happens to be T, which is great because I'm solving for T. So I'm going to take the
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T out of each term and what's left, A minus B.
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Now A minus B, that quantity, is the coefficient of T. So how do I get rid of coefficients?
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I multiply by their multiplicative opposite.
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So 1 over A minus B. So here I have 1 over A minus B on this side.
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So A minus B is in the denominator. And I'm left with T by itself on the right.
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And then I wrote it in line here, just in case you're going to put it in the calculator.
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So I can think about some numbers.
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Say, Q is 10, C was 2, A was 5, and B was 3.
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And instead of solving this thing, instead of putting them in there and solving it, I can
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just use my formula right here and find the answer 6.
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So is it true?
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I'll go back to my original equation and I'm going to plug in all my givens, Q, A, B,
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and C. Now I'm going to plug in the answer that I got for T, which was 6.
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And if I do the arithmetic on this side, I discover that I discover that, let's see,
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did I get everything right in there?
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Yes.
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I discover that I get 10 and 10 is equal to 10.
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So it's true.
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So the answer was 6.
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The answer was 6.
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So I must have solved my literal equation successfully because when I tried an example
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and then back checked it, it came out to be true.
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Okay.
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So we could solve this thing for whatever we wanted.
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We could solve it for A, for example.
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A.
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There's only one term with A in here.
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So everything else would have to go to the other side.
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If I wanted to solve that for A, I would go Q.
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What would I do?
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Q plus C.
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And I have minus BT, so it would be plus BT, is equal to what?
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A T. And now if I'm solving for A, I'd have to divide both sides by T. So I get A is
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equal to Q plus C plus BT, all divided by T. Okay.
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So there's an example of solving that linear literal equation for a parameter, A. Okay.
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In this case, I could easily solve for the parameter B. I could easily solve for the
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parameter C. Then I have a formula for each one of those.
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Then I have a way of finding the value of a parameter if I know the other parameters
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and an input and an output.
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This is beautiful because sometimes that's the case.
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Because I know an input, output pair, I know all the other parameters.
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I'm just missing the value of one of them.
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This allows me to find the value of that one, given that I had an output, an input, and
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the other two parameters, I could find out what the missing parameter value was.
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So can we do this all in the calculator?
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So it's totally slick and fast.
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Yes we can.
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You can do it with this amazing thing called solver.
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Solver.
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So how to solve or work?
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How to solve or work?
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Solver starts with a literal equation, like this one right here.
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That literal equation.
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So let me just rewrite it up here.
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Q is equal to A T minus B T minus C. Okay.
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So before I was talking about zeros and I did a video where we did zeros.
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And how do you do a zero in the calculator?
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You have to get the Q to the other side so that the equation equals zero, equals zero.
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And then when you solve, you're trying to find the value that creates the output of zero.
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It's called a zero.
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So it turns out the solver routine in your calculator needs to solve for a zero.
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So we have to make a little manipulation here.
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We have to manipulate things just a tiny little bit, which is we have to.
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We have to move our Q. We have to move our Q to the other side.
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We need this to be zero.
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And we need to put Q over here so it's minus Q.
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So if we do that, then we're all ready to put it in the calculator.
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So we now have a function whose output is going to be zero when we're solving it.
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Okay.
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So we're going to take this guy.
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We're going to think of it as a function whose output is going to be zero.
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And we're going to put it in y equals.
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We're going to put it in y equals.
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So we're going to put it in, for example, y1.
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Okay.
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So now I'm going to jump to the calculator.
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All right.
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Then jump to the calculator and put this guy in.
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So here I come with a calculator.
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All right.
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Can I see that well enough?
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Yes.
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Okay.
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So what am I going to do here?
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Let me get the key press up here so you can see it.
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So I'm going to press the y equals key.
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And anywhere in here, it doesn't matter.
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Maybe I'm not going to put it in y1 because I want to keep this for later use and I want
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to keep y1 open for when I'm doing other problems.
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So maybe I'll put it all the way down here at y4.
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And this one you can just totally ignore.
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That's another one.
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Ooh.
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That's for calculus class right there.
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So we know we're going to pay attention to that one.
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But let's put this guy in.
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This is I have it.
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So I'm going to put in with alpha.
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Alpha.
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Alpha key here.
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You see it?
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Alpha.
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A.
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So letters I have there.
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Alpha t.
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So you do this with me.
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You've got to find the t.
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T is right above what?
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The four.
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Alpha t minus what do we have left?
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Minus parentheses.
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Alpha b.
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And then alpha t again.
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Alpha t plus c.
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And I'm going to do plus c because I made an error in there, didn't I?
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And then I'm going to do minus alpha.
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And I'm going to go find the q key.
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Where's the q key?
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q is right above the nine.
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Okay.
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00:15:53.240 --> 00:15:57.120
So I put in plus there, but I made a mistake, didn't I?
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That's right there, you see?
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This isn't right right here.
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That should be a plus.
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I transcribed it incorrectly.
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So let me fix that.
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Okay, here I go back to my character.
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Okay, so now I don't have to look there.
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So I got it all in there, all right?
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Now you pay an attention.
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I got it all in there just like I had on the other one.
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If you need to put the video back and make sure you got it in there correctly.
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Now what I'm going to do is I'm going to bring up solver.
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Before I bring up solver actually, I'm going to come back over here and I'm going to turn
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off the way for equals.
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I'm going to turn that off so it doesn't try to plot.
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All right, so I want you to pay attention because I'm going to press some new keys that
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you've never seen before, all right?
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I'm going to do the math key.
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You see it right here?
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The math key.
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I'm going to press that.
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Now, the one that I want is at the very bottom.
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One way to get to the bottom of a list quickly is to go up.
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This is the one I want.
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Solver.
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It's zero.
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If you remember zero, you can just type a zero.
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If you don't, you have to press enter.
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Okay, so when you press enter, there could be any number of things going on in here, okay?
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Any number of things in here.
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So what we need to do is we need to go back up to the very top, okay?
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Back up to the very top and we need to clear it out.
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So when you get in the solver, press the up arrow key until you see this.
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Do you see equation solver, EQN, see that zero?
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Remember it's zero?
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On the left hand side, just like we had.
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Well, that's because that's what solver expects.
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So now we're going to associate that with where we put our function.
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So where did we put our function?
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Let me just check.
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We put it at y4, right?
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We put it at y4.
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So let me go back.
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How do I go back there?
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Math?
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Zero is solver.
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Scroll up.
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Clear.
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Now pay attention.
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How do I get y4 in there?
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I'm going to press the bars key.
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Go over to Y bars.
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Now I'm going to press one to function.
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00:18:22.000 --> 00:18:28.160
You've done this before for other stuff, for evaluation, for example, in the evaluation
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video.
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Now I'm going to go down to 4, press 4, press enter.
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And now when I press enter, again, up comes that literal equation.
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And now I can double check because the letters should match up the letters that I had typed
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into Y4.
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So I do ATBC and Q.
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Now I can jump back to that problem that I just had, right?
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00:18:51.720 --> 00:18:53.880
Because I should get the same answer.
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You remember the answer was 6, right?
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The answer was 6.
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So what were the values of these guys?
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Let's go find them.
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What were the values of the ones that I put in?
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They were Q was 10, C was 2, A was 5.
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Q was 10, C was 2, A was 5.
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I'm going to put 5 and for A. We don't know what T is.
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B was 3, I believe.
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C was what?
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00:19:24.600 --> 00:19:29.960
2, and Q was 10.
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00:19:29.960 --> 00:19:31.960
So let me double check.
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00:19:31.960 --> 00:19:34.760
10, 2, 5 and 3.
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10, 2, 5 and 3.
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I got it.
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00:19:37.000 --> 00:19:38.000
So great.
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Those are the ones that are known.
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T is the one that is unknown.
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00:19:42.200 --> 00:19:43.200
Unknown.
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So I put the cursor on the line of the unknown.
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And that could be an unknown input in this case.
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00:19:50.040 --> 00:19:52.600
Or it could be an unknown parameter.
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00:19:52.600 --> 00:19:55.240
It depends on which one is missing.
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00:19:55.240 --> 00:19:58.520
In solver, you can only do this if you're
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00:19:58.520 --> 00:20:01.520
no one unknown thing, one unknown thing.
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00:20:01.520 --> 00:20:03.960
If you have two unknowns, you can't use solver.
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00:20:03.960 --> 00:20:07.120
But in this case, of the five, we know four of them.
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00:20:07.120 --> 00:20:08.920
That's all that matters.
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00:20:08.920 --> 00:20:12.360
So I'm solving for T. So here's how you make it solve.
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Here's the cool part.
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00:20:14.200 --> 00:20:22.080
You alpha, enter, and boom, up comes 6.
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00:20:22.080 --> 00:20:23.320
OK?
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00:20:23.320 --> 00:20:26.280
So for example, what if I wanted to do this?
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00:20:26.280 --> 00:20:29.880
And I wanted to change the output to 12.
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00:20:29.880 --> 00:20:31.320
Then I could come back and discover
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00:20:31.320 --> 00:20:33.200
that the input is what?
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00:20:33.200 --> 00:20:36.800
Alpha, enter.
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00:20:36.800 --> 00:20:40.160
Alpha, excuse me, the input is 7.
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00:20:40.160 --> 00:20:41.880
So you see the power of this?
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00:20:41.880 --> 00:20:45.440
Once you get it set up, you can do multiple solutions
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00:20:45.440 --> 00:20:46.480
at a time.
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00:20:46.480 --> 00:20:47.600
Just keep doing it.
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00:20:47.600 --> 00:20:48.840
It's easy.
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00:20:48.840 --> 00:20:49.840
OK?
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00:20:49.840 --> 00:20:51.800
So this is the story of solver.
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00:20:51.800 --> 00:20:54.480
And this case, solver being used
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00:20:54.480 --> 00:21:01.400
with a literal linear equation, literal linear equation.
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00:21:01.400 --> 00:21:01.920
OK.
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00:21:05.480 --> 00:21:10.120
So that completes this video that goes with 2.5
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on solving with linear equations.