WEBVTT
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We're going to solve a problem using t variables.
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Ben buys two hats and five scarves in a store for $111.
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Jose buys four hats and three scarves in the same store for $103.
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State the cost of a hat and the cost of a scarf.
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Okay, the clue here is that we're looking for the cost of a hat and a cost of a scarf.
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So let's define our variables by the cost of a hat and the cost of a scarf.
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Cost of a scarf. How about h?
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We'll be the cost of a hat.
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And how about s?
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We'll be the cost of a scarf.
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I think I'll change that to x because some people might think my s is a five.
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So how about x for the cost of a scarf?
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All right, first let's see what happens to Ben here.
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He buys two hats and five scarves.
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All right, so he buys two hats.
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Each hat costs $8.
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So he pays $2h total for the two hats.
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Plus he also has to pay for the scarves.
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He bought five scarves.
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Each one costs $x, right?
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Or we don't know if it's...
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Well, we're going to see that's in dollars, but it could be dollars and cents.
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So that could be five x.
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And the total amount he pays is $111.
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So this first equation tells you what Ben pays.
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Now let's look at the second equation.
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Jose.
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Jose buys four hats and three scarves.
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So he pays four h for the hats, right?
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Four hats, h dollars each.
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And he buys three scarves.
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So we have to add the cost of the scarves, three,
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times the cost of each scarf is x.
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And that's $103.
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All right, not easy to solve for h or x.
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So the substitution method doesn't look like the way to go.
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And I certainly want to have to do this by graphing.
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So I'm going to go ahead and use the addition method.
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I'm going to move this over a little to the right so there's some space.
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Okay.
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So let's see.
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We would like to either have the h's or the x's cancel out.
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So notice 2 goes in to 4.
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So if I just multiply the top equation by negative 2,
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I'm going to get it.
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In fact, I'm going to multiply the top equation by positive 2
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and the bottom equation by negative 1.
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And I'll tell you why I'm doing that.
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Because I know when I multiply 2 times 111,
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it'll make it positive and then I'll be subtracting the 103.
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If you had to multiply it by negative 2,
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you'd have the negative number on top.
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Either way, you will end up with the same thing.
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So you might do it a little bit differently than me.
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Because there are several ways to multiply the equations by a number
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to get the x's or y's to cancel out.
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So I'm going to do 4h plus 10x equals 2, 22.
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And negative 4h, remember you have to multiply both sides of the equation
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every single term by negative 1.
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So it's a negative 4h minus 3x equals negative 103.
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So let me just state this again. If I would have multiplied the top by negative 2,
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all the negatives would be in the top equation and all the ones at the bottom
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would still be positive.
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When you add these two together,
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though, this way I have a positive in front of the x.
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So I have 7x equals 119.
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Let me just divide by 7.
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So x is going to be 17.
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What did x stand for?
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Oh yeah, the cost of the scarf.
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So the cost of the scarf looks like it's going to be $17.
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You know, I need to find the cost of a hat.
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So I want to take either of these two equations.
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I'll take the top one, 2x plus 5x equals 11,
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2h plus 5x equals 11.
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And I'm going to plug in 17 for x.
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So 2h plus 5 times 17 equals 11.
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And that way I'll be able to solve for h and know the cost of a hat.
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So I have 2h plus 85 equals 11.
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And we can subtract 85 from both sides.
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And that gives you 2h equals 26.
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Divide both sides by 2.
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So h is 13.
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So the cost of a hat must be $13.
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If we did this right.
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And that's what we're looking for.
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The cost of one hat and the cost of a scarf.
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All right. Now, you would have gotten the same answer if you have plugged in 17
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into the second equation.
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The 4h plus 3x equals 103.
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All right. So now I want to make sure these answers make sense given the original problem.
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So here's the original problem.
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Here's what we got for the cost of a hat and the cost of a scarf.
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So we're hoping that this is going to be the correct answer right here.
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H and X are not part of the problem.
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You're the one who said just called the cost of a hat H.
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So the cost of a hat is $13.
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Cost of a scarf is $17.
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Now let's check and make sure that's true.
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So originally he said 2 hats and 5 scars.
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Okay. 2 hats.
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All right. How much did he pay for a hat? $13.
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So he paid $26 for a hat.
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And let's see. What else did he buy? 5 scarves.
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Each scarf was $17.
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And so he paid $85 for the scarves.
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And so he paid $111.
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And yes, that checks out for ban.
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Right. So ban checks out.
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Now let's check.
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Jose. Jose's cost.
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He bought 4 hats. Okay. Each hat was $13.
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So he paid $52 for the hats.
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And he bought 3 scarves.
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Each scarf was $17.
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So he paid $51 for a scarf.
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And if we add that up, let's see.
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We get $103.
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And yes, it does state that he spent $103.
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So it checks out that that was the correct amount that Ben would have spent.
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Correct amount. Jose would have spent.
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So therefore this is our answer.
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The cost of a hat is $13.
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And the cost of a scarf is $17.
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Here's what you could do.
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Write this problem down on your own piece of paper.
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Way to half an hour.
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And do it all on your own.
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That's the answer.