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Solving Multi-step Equations

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Hi, my name is Anthony Varela, and in this tutorial, we're going to solve multi-step equations. So we're going to talk about how to isolate the variables so we can solve our equation. We'll talk about inverse operations that will help us isolate the variable. And we'll talk about the order of operations. What order are we going to apply those inverse operations to solve our equations?

So first, let's review general guidelines for solving an equation. First, we're going to isolate the variable using inverse operations. So the inverse operations we're going to talk about today are addition and subtraction, multiplication and division, and powers and roots. And then another important piece when solving multi-step equations is that we're going to perform these inverse operations using the reverse order of operations that we first encountered when we were learning how to evaluate expressions. So solving an equation uses inverse operations and reverse order of operations.

And of course, when you're solving an equation, whatever you do to one side, you have to do to the other. We call that the rule of equality. So that fills up our notes for today's lesson. Let's get into some examples.

This example involves distribution, where we have an outside factor and then we have a sum in parentheses, there. And there are two main ways you can go about solving this equation. One thing you could do is go ahead and show that distribution. So you could say that this is negative 6x plus 10 equals 25. Just multiplying negative 3x plus 5, each of those terms, by 2.

Another thing that you could do is divide both sides of the equation by 2 right away. And you'll get negative 3x plus 5 equals 12 and 1/2. So you're simplifying the equation before you even begin to solve for it. And you can do any which way you like, whatever you prefer. Let's go ahead and solve both of these equations and we should get the same solution. So let's focus on the one that we see on our left.

Now, the operations that I see. Well, I see x being multiplied by negative 6, and then I see that plus 10. And I'm going to use inverse operations, so I'm going to be using some division and I'm going to be using some subtraction. But I apply these inverse operations in reverse order of operations, so the order of operations when I was evaluating an expression told me to perform multiplication and division before addition and subtraction.

Well, here, I'm going to apply subtraction first. So I'm going to subtract 10 from both sides of my equation. Now, this, what it does is it moves that plus 10 over to the other side of the equation as our negative 10. So we have just negative 6x equals 15.

Now I can apply my inverse operation for multiplication, which is division, so dividing both sides of the equation by negative 6. So I get just x equals negative 2.5. Now let's go ahead and solve the equation we see on our right. And we should end up with x equals negative 2.5.

Once again, the inverse operations that I'm going to apply here are division and subtraction, but I'm doing subtraction first, so subtracting 5 from both sides of the equation. This gives me negative 3x equals 7.5. Now I can divide both sides by that negative 3. This will isolate x, and x equals negative 2.5.

Our next example involves a variable in the denominator. So we see nine equals 6 over x. X is our variable, and it's in the denominator of this fraction. So what you want to do if you ever see a variable in the denominator is you want to get it out of the denominator.

How do we do that? Well, we're going to multiply both sides of our equation by whatever we see in the denominator. So that would be multiplying 6 over x by x. So now that's going to cancel out what we have in the denominator, but we have to multiply the other side of the equation by x as well. Whatever we do to one side, we have to do to the other.

So now what does our equation look like? We have 9x equals 6, which is something we're probably familiar with. So now we can divide both sides by 9, and we get that x equals 6/9, which can be simplified to 2/3.

The next example that I'd like to go through involves a radical. So we have some expression, 4x plus 17, and it's underneath a square root. So we're taking the square root of this entire quantity, 4x plus 17, and that is equal to 3. How are we going to isolate x? And this is a little bit tricky, because we're looking at operations that we have where we have a radical, we have multiplication with the 4x, and we have addition with the plus 17.

And so thinking about the order of operations, if we're evaluating an expression, we would evaluate anything that's the square root first. But remember that with radicals, there is implied grouping symbols as well. So we can imagine that there's parentheses around this expression. And in order of operations, we evaluate things in parentheses first.

So thinking about the reverse order of operations, we're going to leave that alone for now, and we're going to cancel out the radical. Now, how do we cancel out the radical? The inverse operation of the square root is to square entire quantities. And whatever we do to one side of the equation, we have to do to the other. So we undid the radical on the left side of the equation, and now we just need to right in 3 squared, which is 9.

And this looks like something we can solve. What we're going to do first is the inverse operation of addition, which is subtraction. So take away 17 from both sides. 4x equals negative 8. And our last step is to divide both sides by 4, and we get that x equals negative 2.

So let's review our notes for solving multi-step equations. Remember, inverse operations isolates our variable. Our inverse operations that we talked about today are addition and subtraction, multiplication and division, and powers and roots. And we use inverse operations following our reverse order of operations to isolate our variable. Well, that's it for solving multi-step equations. Thanks for watching, and hope to see you next time.