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2nd SESSION part 1: What is an equation?

 TRANSCRIPTION

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Hi, I’m Rob. Welcome to Math Antics.
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In our last Algebra video, we learned that Algebra involves equations
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that have variables or unknown values in them.
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And we learned that solving an equation means figuring out what those unknown values are.
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In this video, we’re going to learn how to solve some very simple
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Algebraic equations that just involve addition and subtraction.
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Then in the next video, we’ll learn how to solve some simple equations involving multiplication and division.
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Are you ready?… I thought so!
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Okay… so if you’ve got an equation that has an unknown value in it,
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then the key strategy for solving it is to rearrange the equation
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until you have the unknown value all by itself on one side of the equal sign,
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and all of the known numbers on the other side of the equal sign.
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Then, you’ll know just what the unknown value is.
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But, how do we do that? How do we rearrange equations?
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Well, we know that Algebra still uses the four main arithmetic operations
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(addition, subtraction, multiplication and division)
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and we can use those operations to rearrange equations,
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as long as we understand one really important thing first.
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We need to understand that an equation is like a balance scale.
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You’ve seen a balance scale, right?
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If there’s the same amount of weight on each side of the scale,
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then the two sides are in balance.
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But, if we add some weight to one side...
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then the scale will tip.
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The two sides are no longer in balance.
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An equation is like that.
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Whatever is on one side of the equal sign MUST have exactly the same value
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as whatever is on the other side.
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Otherwise, the equation would not be true.
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Of course, that doesn’t mean that the two sides have to look the same.
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For example, in the equation 1 + 1 = 2,
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1 + 1 doesn’t LOOK the like the number 2,
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but we know that 1 + 1 has the same VALUE as 2,
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so 1 + 1 = 2 is in balance. It’s a true equation.
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The reason we need to know that equations must be balanced
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is because when we start rearranging them, if we are not careful,
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we might do something that would change one of the sides more than the other.
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That would make the equation get out of balance and it wouldn’t be true anymore.
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And if that happens, we won’t get the right answer when we solve it.
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That sounds pretty bad, huh? So how do we avoid that?
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How do we avoid getting an equation out of balance?
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The key is that whenever we make a change to an equation,
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we have to make the exact same change on both sides
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That’s so important, I’ll say it again.
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Whenever we do something to an equation,
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we have to do the same thing to BOTH sides.
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For example, if we want to add something to one side of an equation,
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we have to add that same thing to the other side.
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And if we want to subtract something from one side of an equation,
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then we have to subtract that same thing from the other side.
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And it’s the same for multiplication and division.
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If we want to multiply one side of an equation by a number,
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then we need to multiply the other side by that same number.
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Or if we want to divide one side of an equation by a number,
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then we have to divide the other side by that number also.
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As long as you always do the same thing to both sides of an equation,
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it will stay in balance and your equation will still be true.
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Alright, like I said, in this video, we’re just going to focus on equations involving addition and subtraction.
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And here’s our first example: x + 7 = 15
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To solve for the unknown value ‘x’,
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we need to rearrange the equation so that the ‘x’ is all by itself on one side of the equal sign.
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But what can we do to get ‘x’ all by itself?
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Well, right now ‘x’ is not by itself because 7 is being added to it.
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Is there a way for us to get rid of that 7?
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Yes! Since seven is being added to the ‘x’, we can undo that by subtracting 7 from that side of the equation.
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Subtracting 7 would leave ‘x’ all by itself because ‘x’ plus 7 minus 7 is just ‘x’.
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The ‘plus 7’ and the ‘minus 7’ cancel each other out.
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Okay great! So we just subtract 7 from this side of the equation and ‘x’ is all by itself.
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…equation solved, right?
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WRONG! If we just subtract 7 from one side of the equation and not the other side,
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then our equation won’t be in balance anymore.
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To keep our equation in balance, we also need to subtract 7 from the other side of the equation.
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But on that side, we just have the number 15.
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So we need to subtract 7 from that 15.
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And since 15 - 7 = 8, that side of the equation will just become 8.
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There, by subtracting 7 from BOTH sides, we’ve changed the original equation (x + 7 = 15)
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into the new and much simpler equation (x = 8) which tells us that the unknown number is 8.
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We have solved the equation!
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And to check our answer, to make sure we got it right,
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we can see what would happen if we replaced the unknown value in our original equation with the number 8.
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Instead of x + 7 = 15, we’d right 8 + 7 = 15,
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and if that’s true, then we know we got the right answer.
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Pretty cool, huh?
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Let’s try another one: 40 = 25 + x
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This time, the unknown value is on the right hand side of the equation.
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Does that make it harder?
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Nope. We use the exact same strategy.
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We want to get ‘x’ by itself, but this time ‘x’ is being added to 25.
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But thanks to the commutative property, that’s the same as 25 being added to ‘x’.
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So, to isolate 'x', we should subtract 25 from that side of the equation.
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But then we also need to subtract 25 from the other side to keep things in balance.
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On the right side, x plus 25 minus 25 is just x
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The minus 25 cancels out the positive 25 that was there.
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And on the other side we have 40 minus 25 which would leave 15.
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So the equation has become 15 = x, which is the same as x = 15.
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Again, we’ve solved the equation.
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So, whenever something is being added to an unknown,
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we can undo that and get the unknown all by itself
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by subtracting that same something from both sides of the equation.
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But what about when something is being subtracted from an unknown,
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like in this example: x - 5 = 16
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In this case, ‘x’ is not by itself because 5 is being subtracter or taken away from it.
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…any ideas about how we could get rid of (or undo) that ‘minus 5’?
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Yep! To undo that subtraction, this time we need to ADD 5 to both sides of the equation.
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The ‘minus 5’ and the ‘plus 5’ cancel each other out and leave ‘x’ all by itself on this side.
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And on the other side, we have 16 + 5 which is 21.
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So in this equation, x equals 21.
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Let’s try another example like that: 10 = x - 32.
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Again the ‘x’ is not by itself because 32 is being subtracted from it.
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So to cancel that ‘minus 32’ out, we can just add 32 to both sides of the equation.
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On the right side, the ‘minus 32’ and the ‘plus 32’ cancel out leaving just ‘x’.
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And on the left we have 10 + 32 which is 42. Now we know that x = 42.
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Okay, so now you know how to solve very simple equations like these
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where something is being added to an unknown or where something is being subtracted from an unknown.
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But before you try practicing on your own,
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I want to show you a tricky variation of the subtraction problem
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that confuses a lot of students.
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Do you remember how subtraction does NOT have the commutative property?
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If you switch the order of the subtraction, it’s a different problem.
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Suppose we get a problem, where instead of a number being taken away from an unknown,
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an unknown is being taken away from a number.
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What do we do in that case?
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Well, we still want to get the unknown all by itself,
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but it’s a little harder to see how to do that.
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In this problem (12 - x = 5) the 12 on this side is a positive 12,
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so we could subtract 12 from both sides.
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That would get rid of the 12,
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but the problem is that wouldn’t get rid of this minus sign.
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That’s because the minus sign really belongs to the ‘x’ since it’s the ‘x’ that is being subtracted.
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Subtracting 12 would leave us with ‘negative x’ on this side of the equal sign,
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which is not wrong, but it might be confusing if you don’t know how to work with negative numbers yet.
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Fortunately, there’s another way to do this kind of problem that will avoid getting a negative unknown.
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Instead of subtracting 12 from both sides, what would happen if we added ‘x’ to both sides?
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Can we do that? Can we add an unknown to both sides?
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Well sure! why not?
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We can add or subtract ANYTHING we want as long as we do it to both sides!
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And when we do that, the ‘minus x’ and the ‘plus x’ will cancel each other out on this side.
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And and the other side, we will get 5 + x.
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Now our equation is 12 = 5 + x.
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And you might be thinking, “but why would we do that? That didn’t even solve our equation!”
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That’s true, but it changed it into an equation that we already know how to solve.
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Now it’s easy to see that we can isolate the unknown just by subtracting 5 from both sides of the equation.
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That will give us 7 = x or x = 7.
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It just took us one extra step to rearrange the equation,
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but then it was easy to solve.
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Okay, that’s the basics of solving simple algebraic equations that involve addition and subtraction.
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You just need to get the unknown value all by itself,
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and you can do that by adding or subtracting something from both sides of the equation.
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And this process works the same even if the numbers in the equations are decimals or fractions.
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And it also works the same no matter what symbol you are using as an unknown.
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It could be x, y, z or a, b, c.
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The letter being used doesn’t matter.
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Remember, when it comes to math,
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it’s really important to practice what you’ve learned.
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So be sure to try solving some basic equations on your own!
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As always, thanks for watching Math Antics and I’ll see ya next time.

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