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  • Hi, I’m Rob. Welcome to Math Antics.

  • In our last Algebra video, we learned that Algebra involves equations

  • that have variables or unknown values in them.

  • And we learned that solving an equation means figuring out what those unknown values are.

  • In this video, were going to learn how to solve some very simple

  • Algebraic equations that just involve addition and subtraction.

  • Then in the next video, well learn how to solve some simple equations involving multiplication and division.

  • Are you ready?… I thought so!

  • Okayso if youve got an equation that has an unknown value in it,

  • then the key strategy for solving it is to rearrange the equation

  • until you have the unknown value all by itself on one side of the equal sign,

  • and all of the known numbers on the other side of the equal sign.

  • Then, youll know just what the unknown value is.

  • But, how do we do that? How do we rearrange equations?

  • Well, we know that Algebra still uses the four main arithmetic operations

  • (addition, subtraction, multiplication and division)

  • and we can use those operations to rearrange equations,

  • as long as we understand one really important thing first.

  • We need to understand that an equation is like a balance scale.

  • Youve seen a balance scale, right?

  • If there’s the same amount of weight on each side of the scale,

  • then the two sides are in balance.

  • But, if we add some weight to one side...

  • then the scale will tip.

  • The two sides are no longer in balance.

  • An equation is like that.

  • Whatever is on one side of the equal sign MUST have exactly the same value

  • as whatever is on the other side.

  • Otherwise, the equation would not be true.

  • Of course, that doesn’t mean that the two sides have to look the same.

  • For example, in the equation 1 + 1 = 2,

  • 1 + 1 doesn’t LOOK the like the number 2,

  • but we know that 1 + 1 has the same VALUE as 2,

  • so 1 + 1 = 2 is in balance. It’s a true equation.

  • The reason we need to know that equations must be balanced

  • is because when we start rearranging them, if we are not careful,

  • we might do something that would change one of the sides more than the other.

  • That would make the equation get out of balance and it wouldn’t be true anymore.

  • And if that happens, we won’t get the right answer when we solve it.

  • That sounds pretty bad, huh? So how do we avoid that?

  • How do we avoid getting an equation out of balance?

  • The key is that whenever we make a change to an equation,

  • we have to make the exact same change on both sides

  • That’s so important, I’ll say it again.

  • Whenever we do something to an equation,

  • we have to do the same thing to BOTH sides.

  • For example, if we want to add something to one side of an equation,

  • we have to add that same thing to the other side.

  • And if we want to subtract something from one side of an equation,

  • then we have to subtract that same thing from the other side.

  • And it’s the same for multiplication and division.

  • If we want to multiply one side of an equation by a number,

  • then we need to multiply the other side by that same number.

  • Or if we want to divide one side of an equation by a number,

  • then we have to divide the other side by that number also.

  • As long as you always do the same thing to both sides of an equation,

  • it will stay in balance and your equation will still be true.

  • Alright, like I said, in this video, were just going to focus on equations involving addition and subtraction.

  • And here’s our first example: x + 7 = 15

  • To solve for the unknown value ‘x’,

  • we need to rearrange the equation so that the ‘x’ is all by itself on one side of the equal sign.

  • But what can we do to get ‘x’ all by itself?

  • Well, right now ‘x’ is not by itself because 7 is being added to it.

  • Is there a way for us to get rid of that 7?

  • Yes! Since seven is being added to the ‘x’, we can undo that by subtracting 7 from that side of the equation.

  • Subtracting 7 would leave ‘x’ all by itself because ‘x’ plus 7 minus 7 is just ‘x’.

  • Theplus 7’ and theminus 7’ cancel each other out.

  • Okay great! So we just subtract 7 from this side of the equation and ‘x’ is all by itself.

  • equation solved, right?

  • WRONG! If we just subtract 7 from one side of the equation and not the other side,

  • then our equation won’t be in balance anymore.

  • To keep our equation in balance, we also need to subtract 7 from the other side of the equation.

  • But on that side, we just have the number 15.

  • So we need to subtract 7 from that 15.

  • And since 15 - 7 = 8, that side of the equation will just become 8.

  • There, by subtracting 7 from BOTH sides, weve changed the original equation (x + 7 = 15)

  • into the new and much simpler equation (x = 8) which tells us that the unknown number is 8.

  • We have solved the equation!

  • And to check our answer, to make sure we got it right,

  • we can see what would happen if we replaced the unknown value in our original equation with the number 8.

  • Instead of x + 7 = 15, we’d right 8 + 7 = 15,

  • and if that’s true, then we know we got the right answer.

  • Pretty cool, huh?

  • Let’s try another one: 40 = 25 + x

  • This time, the unknown value is on the right hand side of the equation.

  • Does that make it harder?

  • Nope. We use the exact same strategy.

  • We want to get ‘x’ by itself, but this time ‘x’ is being added to 25.

  • But thanks to the commutative property, that’s the same as 25 being added to ‘x’.

  • So, to isolate 'x', we should subtract 25 from that side of the equation.

  • But then we also need to subtract 25 from the other side to keep things in balance.

  • On the right side, x plus 25 minus 25 is just x

  • The minus 25 cancels out the positive 25 that was there.

  • And on the other side we have 40 minus 25 which would leave 15.

  • So the equation has become 15 = x, which is the same as x = 15.

  • Again, weve solved the equation.

  • So, whenever something is being added to an unknown,

  • we can undo that and get the unknown all by itself

  • by subtracting that same something from both sides of the equation.

  • But what about when something is being subtracted from an unknown,

  • like in this example: x - 5 = 16

  • In this case, ‘x’ is not by itself because 5 is being subtracter or taken away from it.

  • any ideas about how we could get rid of (or undo) thatminus 5’?

  • Yep! To undo that subtraction, this time we need to ADD 5 to both sides of the equation.

  • Theminus 5’ and theplus 5’ cancel each other out and leave ‘x’ all by itself on this side.

  • And on the other side, we have 16 + 5 which is 21.

  • So in this equation, x equals 21.

  • Let’s try another example like that: 10 = x - 32.

  • Again the ‘x’ is not by itself because 32 is being subtracted from it.

  • So to cancel thatminus 32’ out, we can just add 32 to both sides of the equation.

  • On the right side, theminus 32’ and theplus 32’ cancel out leaving just ‘x’.

  • And on the left we have 10 + 32 which is 42. Now we know that x = 42.

  • Okay, so now you know how to solve very simple equations like these

  • where something is being added to an unknown or where something is being subtracted from an unknown.

  • But before you try practicing on your own,

  • I want to show you a tricky variation of the subtraction problem

  • that confuses a lot of students.

  • Do you remember how subtraction does NOT have the commutative property?

  • If you switch the order of the subtraction, it’s a different problem.

  • Suppose we get a problem, where instead of a number being taken away from an unknown,

  • an unknown is being taken away from a number.

  • What do we do in that case?

  • Well, we still want to get the unknown all by itself,

  • but it’s a little harder to see how to do that.

  • In this problem (12 - x = 5) the 12 on this side is a positive 12,

  • so we could subtract 12 from both sides.

  • That would get rid of the 12,

  • but the problem is that wouldn’t get rid of this minus sign.

  • That’s because the minus sign really belongs to the ‘x’ since it’s the ‘x’ that is being subtracted.

  • Subtracting 12 would leave us withnegative x’ on this side of the equal sign,

  • which is not wrong, but it might be confusing if you don’t know how to work with negative numbers yet.

  • Fortunately, there’s another way to do this kind of problem that will avoid getting a negative unknown.

  • Instead of subtracting 12 from both sides, what would happen if we added ‘x’ to both sides?

  • Can we do that? Can we add an unknown to both sides?

  • Well sure! why not?

  • We can add or subtract ANYTHING we want as long as we do it to both sides!

  • And when we do that, theminus x’ and theplus x’ will cancel each other out on this side.

  • And and the other side, we will get 5 + x.

  • Now our equation is 12 = 5 + x.

  • And you might be thinking, “but why would we do that? That didn’t even solve our equation!”

  • That’s true, but it changed it into an equation that we already know how to solve.

  • Now it’s easy to see that we can isolate the unknown just by subtracting 5 from both sides of the equation.

  • That will give us 7 = x or x = 7.

  • It just took us one extra step to rearrange the equation,

  • but then it was easy to solve.

  • Okay, that’s the basics of solving simple algebraic equations that involve addition and subtraction.

  • You just need to get the unknown value all by itself,

  • and you can do that by adding or subtracting something from both sides of the equation.

  • And this process works the same even if the numbers in the equations are decimals or fractions.

  • And it also works the same no matter what symbol you are using as an unknown.

  • It could be x, y, z or a, b, c.

  • The letter being used doesn’t matter.

  • Remember, when it comes to math,

  • it’s really important to practice what youve learned.

  • So be sure to try solving some basic equations on your own!

  • As always, thanks for watching Math Antics and I’ll see ya next time.

  • Learn more at www.mathantics.com

Hi, I’m Rob. Welcome to Math Antics.

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Algebra Basics: Solving Basic Equations Part 1 - Math Antics

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    Yassion Liu   posted on 2016/07/22
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