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Solving Linear Equations · Acedra
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Solving Linear Equations
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0:00
Let's try some slightly more complicated equations. Let's say we have 3
0:13
times x plus 5 is equal to 17.
0:25
is equal to 17. So what's different about this and what we saw in the
0:29
last video is all of a sudden now we have this plus 5. If it
0:32
was just 3x is equal to 17, you could divide both sides by 3 and
0:36
you'd get your answer. But now this 5 seems to mess things up a little
0:40
bit. Now before we even solve it, let's think about what it's saying. Let's solve
0:44
it kind of in a tangible way and then we'll solve it using operations that
0:49
hopefully will make sense after that. So 3 times x literally means, so let me
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write it over here, so we have 3 times So you literally have an x
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plus an x plus an x. That right there is a 3x. And then that's
1:02
plus 5. And I'm actually going to write it out as 5 objects. So plus
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1 2 3 4 5.
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This right here is 3x plus 5 is equal to 17. So
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let me write the equal sign is equal to 17. Now let me draw 17
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objects here. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
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13 14 15 16 17.
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Now, these two things are equal. So anything you do to this side, you'd have
1:41
to do to that side. If we were to get rid of one object here,
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you'd want to get rid of one object there in order for the equality to
1:46
still be true. Now, what can we do to both sides of this equation so
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we can get it in the form that we're used to, where we only have
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a 3x on the left-hand side, where we don't have this 5? Well, ideally, we
1:59
would just get rid of these five objects here. you would literally get rid of
2:03
these five objects. One, two, three, four, five. But like I said, if the original
2:08
thing was equal to the original thing on the right, if we get rid of
2:11
five objects from the left-hand side, we have to get rid of five objects from
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the right-hand side. So we have to do it here, too. One, two,
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three, four, five. What is a symbolic way of representing taking away 5 things? Well,
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you're subtracting 5 from both sides of this equation. So that's what we're doing here
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when we took away 5 from the left and from the right. So we're subtracting
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5 from the left. That's what we did here. And we're also subtracting 5 from
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the right. Do that right over there. Now, what does the left-hand side of the
2:43
equation now become? The left-hand side, you have 5 minus 5. These cancel out. You're
2:48
just left with the 3x. You are
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just left with the 3x. The 5 and the negative 5 canceled
3:00
out. And you see that here. When you got rid of these five objects, we
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were just left with the 3x's. This right here is the 3x. And the whole
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reason why we subtracted 5 is because we wanted this 5 to go away. Now
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what does the right-hand side of the equation look like? So 3x is going to
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be, let me write the equality sign right under it. is equal to, well you
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can either just do it mathematically, say OK, 17 minus 5. 17 minus 5 is
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12. Or you could just count over here. I had 17 things. I took away
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5. I have 12 left. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
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11, 12. 12. That's what subtraction
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is. It's just taking away 5 things. So now we have it in a pretty
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straightforward form. 3x is equal to 12. All we have to do is divide both
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sides of this equation by 3. So we're just left with an x on the
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left-hand side. So we divide by, let me pick a nicer color than that. Let
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me do this pink color. So you divide the left-hand side by 3, the right-hand
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side by 3. And remember what that's equivalent to. The left-hand side, none of this
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stuff exists anymore. So we should ignore it. None of this stuff exists anymore. In
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fact, let me clear it out, just so that we don't even have to look
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at it. We subtracted it. So let me clear it.
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Let me clear it over here. And so now we are dividing both sides by
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3. Divide the left-hand side by 3. It's 1, 2, 3. So three groups, each
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of them have an x in it. And if you divide this right-hand side by
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3, let's see, you have 1, 2 and 3. So it's 3 groups of 4.
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So when you do it mathematically here, the 3's cancel out. 3 times something divided
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by 3 is just the something. So you're left with x is equal to, and
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then 12 divided by 3 is 4. You get
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x is equal to 4. And you get that exact same thing over here. When
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you divided 3x into groups of 3, each of the groups had an x in
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it. And we divided 12 into groups of 3. Each of the groups have a
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4 in it. So x must be equal
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to 4. Let's do another one. And this time I won't draw it all out
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like this, but hopefully you'll see that the same type of processes are involved. Let
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me scroll down a little bit. Let's say I have 7x.
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I'll do a slightly more complicated one this time. 7x minus 2 is
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equal to, let's say, is equal
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to, I'll make the numbers not work out nice and clean, is equal to negative
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Now this all of a sudden becomes a lot more, you have a negative sign,
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we have a negative over here, but we're going to do the exact same thing.
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The first thing we want to do if we want to get the left-hand side
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simplified to just 7x is we want to get rid of this negative 2. And
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what can we add or subtract to both sides of the equation to get rid
6:12
of this negative 2? Well, if we add 2 to the left-hand side, these two
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guys will cancel out. But remember, this is equal to that. If we want the
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equalities to hold, if we add 2 to the left-hand side, we also have to
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do it to the right-hand side. So what is the new left-hand side going to
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be equal to? So we have 7x. Negative 2 plus 2 is just 0.
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I could write plus 0. I could just write nothing there, and I'll just write
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nothing. So we get 7x is equal to, now what's negative 10 plus 2. And
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this is a little bit of a review of adding and subtracting negative numbers. Remember,
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I'll draw the number line here for you. If I draw the number line, so
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this is 0, this is 1, we could keep going in the positive direction. Negative
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10 is out here. Negative 10, negative 9, negative 8, negative 7, there's a bunch
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of numbers here. I don't have space to draw them all, but we're starting at
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negative 10, and we're adding 2 to it. So we're moving in the positive direction
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on the number line. So we're going 1, 2. So it's negative 8. Don't get
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confused. Don't say, OK, 10 plus 2 is 12, so negative 10 plus 2 is
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negative 12. No. Negative 10 minus 2 would be negative 12, because you'd be going
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more negative. Here, we have a negative number, but we're going to the right. We're
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going in the positive direction. So this is negative 8. So we have 7x is
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equal to negative 8. So now you might be saying, well, how do I do
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this type of a problem, I have a negative number here. You do it the
7:44
exact same way. If we want to just have an x on the left-hand side,
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we have to divide the left-hand side by 7. So that the 7x divided by
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7, the 7's canceled out, you're left with x. So let's do that. If you
7:58
divide by 7, those cancel out. But you can't just do it to the left-hand
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side. Anything you do to the left, you have to do to the right in
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order for the equality to still hold true. So let's divide. the right by 7
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as well. And we are left with just an x. Just an x is equal
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to negative 8 divided by 7. We could work it out. It'll be some type
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of a decimal if you were to use a calculator. Or you could just leave
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it in fraction form. Negative 8 divided by 7 is negative 8 sevenths. Negative 8
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sevenths, or if you want to write it as a mixed number, x is equal
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to 7 goes into 8 one time and has a remainder of 1. So it's
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negative 1 and 1 seventh. Either one would be acceptable.
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