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Solving Single-Step Equations

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Today we're going to talk about solving a single step equation. An equation is just a mathematical statement that says that two expressions or quantities have the same value.

So we'll go over the properties of an equation. We'll talk about the process for solving an equation, and then we'll do some examples with single step equations.

So let's talk a little bit more about what exactly an equation is. As I said before, an equation is just a statement that says that two expressions or quantities are equal in value. So this equation is telling us that x plus 3 has the same value as 5. So in our equation, we do have a variable x, but in an equation your variable is going to have one value that will make the equation true. And in this case, if x plus 3 has to be equal to 5, then that means that x has to be equal to 2.

And we also have something called the rule of symmetry, which says that we can reverse our equation and it will still hold true. So for example, 5 is also equal to x plus 3. And we can see that's true because our x value-- our variable-- will still be equal to 2 in order for this equation to remain true.

So let's look a little bit at the process for solving an equation.

Let's look at the process for solving an equation. When you're solving an equation, you want to isolate the variable that you're trying to solve. So I want to get x by itself, which means I need to cancel out this plus 6.

To do that I'm going to use an inverse operation. So the inverse operation of adding 6 is subtracting 6. So if I go ahead and subtract 6 on this side, again plus adding 6 and subtracting 6 cancel each other out. So I'm left with just x. I've isolated my variable.

However, we have a rule of equality which tells us that whatever we do on one side of the equation we have to do on the other. And that's to keep our equation balanced or keep the statement true. So if I subtract 6 on this side of the equation, I need to subtract 6 on this side of the equation. So now my statement is that x is equal to 10 minus 6. So this represents my solution for x, and I can simplify that, which would tell me that x is equal to 4.

So whenever you find your solution, you can plug that back in to your original equation to see if the statement still holds true. So looking at my original equation, which was x plus 6 equals 10, if I substitute 4 back into that equation, I see that 4 plus 6 does indeed equal 10. So my solution of 4 was correct. So let's take a look at some other inverse operations that we can use to solve equations.

So let's look a little bit more closely at our inverse operations. So our first pair of inverse operations are addition and subtraction. So these operations will cancel each other out or undo each other. For example, if I have 3 and I add 4 to that, I know that's going to give me 7. But if I take 7 and subtract 4 from that, I'm back to my original value of 3. So we can see that those cancel each other out where they are inverse operations of each other.

Another pair of inverse operators are multiplication and division. For example, if I have 10 times 2 that gives me 20. But if I take 20 and I divide by 2, I'm back to my original value of 10. So again, multiplication and division are inverse operators because they cancel each other out.

Another pair of inverse operators are squaring and square root. If I have 5 squared, that's going to give me 25. But if I take the square root of 25, I'm back to my original value of 5. And we can extend that to nth powers and nth roots, so it can work for any exponent in any radical, any nth root. So for example, if I have 2 to the third power, that's going to give me 8. But if I take the cubed root of 8, that's going to bring me back to my original value of 2.

So let's do some examples solving single-step equations using our inverse operators. Let's do some examples.

So my first example is 7 equals x plus 3. I know that the inverse operation for adding 3 will be subtracting 3, and by the property of equality I need to do that to both sides of my equation. So here, adding 3 and subtracting 3 will cancel out, and I've isolated my x variable. And simplifying 7 minus 3 on the other side, I find that x is equal to 4. I can check it by substituting 4 back into my original equation. 4 plus 3 is equal to 7. So this is correct

For my second example I've got a 4 multiplying by x, so I know that I'm going to divide on both sides-- whoops, divide by 4 on both sides to cancel out the multiplication. So now I've isolated my x variable, and 20 divided by 4 is equal to 5. So plugging 5 back into my original equation, 4 times 5 does equal 20. So this is correct.

Here I've got a fraction. A fraction means dividing, so to cancel out dividing by 3, I'm going to multiply by 3, again, on both sides. So here my 3's are going to cancel out, and I'm left with x. And on the other side, 5 times 3 gives me 15. I can check it-- 15 divided by 3 does give me 5. So this is correct.

So in this equation, I've have x squared, and so I'm going to cancel that out by taking the square root again on both sides. Here they'll cancel, so I've isolated my x variable. And now the square root of 36 is going to give me 6.

But, when you take the square root of a number, you can also get the opposite of your answers, so negative 6 is also equal to x. And we can see that's true by plugging it into our original equation. Positive 6 squared or 6 times 6 does give me 36, and negative 6 squared, or negative 6 times negative 6, also gives me a positive 36. So here I have two solutions.

And lastly I've got square root. I'm going to cancel that out by squaring both sides. So on this side, my square root and my square cancel out, so I'm left with x. And on this side, 7 squared is 49. Again I'll check it. The square root of 49 does in fact give me 7. So this is correct.

So let's go over our key notes from today. We first talked about the fact that solving an equation involves isolating the variable that you're trying to solve for by using an inverse operation. And we talked about the fact that any operation that you do on one side of the equation needs to be done on the other side.

And lastly, we looked at when you're doing your examples, you can always check your solution by substituting it in for the variable in your original equation and seeing if the statement holds true. And that is just a good practice to get into when you're doing simple equations because when you do things more complicated, it's more likely that you're going to be making a mistake.

So I hope these notes and examples helped you understand a little bit about solving single-step equations. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.