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Isolating Variables

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Today we're going to talk about isolating variables. Isolating variables is what you need to do when you want to solve an equation. And to isolate a variable, you need to use the inverse operation or more than one inverse operation.

So we're going to start by reviewing the inverse operations, go over the process for solving an equation, and then do some examples.

So let's look at the general process for solving an equation using a simple example. I've got x plus 3 is equal to 10. So the process for solving an equation involves isolating the variable, as I said before. To do that, we want to use the inverse operation. So as we were just looking at our inverse operations, I know that the inverse of adding a number is subtracting a number, the same number.

So I'm going to go ahead and subtract 3, and this is going to undo my adding operation, or cancel it out. So now I've isolated my x variable on this side of the equation.

However, there is a rule of equality when you're solving an equation. And that is that whatever you do to one side of an equation you have to do it to the other. And that's so that you keep the equation balanced and keep the equality statement true. So since I subtracted 3 on this side, I need to subtract 3 on this side.

And now 10 minus 3 is what my x variable is equal to. So I can simplify this. 10 minus 3 gives me 7, so x is equal to 7. So I've solved my equation and found that x is equal to 7.

I can verify that by substituting 7 back into my original equation. And I can see that this statement is true when x is equal to 7 because 7 plus 3 does equal 10.

So let's also look at our inverse operations. Remember, these are operations that will cancel each other out. So first, addition and subtraction. If I were to start with the value of 5 and add 10 to that, that would give me 15. But if I then again take 15 and subtract 10, that's going to bring me back to my original value of 5. So adding 10 and subtracting 10 cancel each other out.

Multiplication and division are also inverse of each other. If I start with 3 and multiply that by 6, that's going to give me 18. If I then take 18 and divide it by 6, that's going to bring me back to my original value of 3. So multiplication and division cancel each other out.

And finally, powers and radicals. If I start with 4 and take that to the second power, that's going to give me 16. If I then take the square root of 16, that's going to bring me back to my original value of 4. So squaring something and taking the square root will cancel each other out, and this will hold true for any power like 4 to the third would be canceled out by the cubed root of a number.

So let's do some examples using our inverse operations.

So here's my first example. I've got 5 is equal to 3 times some number plus 6. So we're trying to figure out what this number is, and we're going to do that using inverse operations and isolating the x variable.

So because the x variable is being multiplied by 3, I know I'm going to have to use division to undo that multiplication. And we are also adding 6, which means I'm going to have to use the subtraction operation to undo that operation.

However, how do you know which one you're supposed to do first? Well in general, what you do is you move any term to the other side using adding and subtracting, and then you cancel out any coefficient or number being multiplied by the variable or any other operation being done to the variable like a power.

So another way to think about that is by doing order of operations backwards. So if we look at PEM/DA/S, which was our acronym for remembering the order of operations, in general we want to go backwards. So again, starting with adding and subtracting, then multiplying or dividing, than any exponents or parentheses.

So we're going to start with canceling out my addition using subtraction. So I'm going to subtract 6 from both sides. And again, these are going to cancel out. 5 minus 6 is going to give me a negative 1. And I'm left with 3 times x on the other side.

Now I can move onto canceling out my multiplication operation. I'm going to do that by dividing. And I need to do it again on both sides. So 3 divided by 3, that will cancel out, and I've isolated my x variable.

And now I can either leave this as a fraction, negative 1 over 3, or I can convert it into a decimal, which is negative 0.3 repeating. So this is the value for my x.

So here's my second example. I've got 4 times 2x minus 10 in parentheses is equal to 20. So this equation needs to be simplified first because we can use the distributive property to multiply the four times the 2x and four times the negative 10 or minus 10.

So I made a note here that before you can use your inverse operations by using PEM/DA/S backwards, you want to make sure that you simplify if you can.

So I'm going to distribute the 4 to the 2x, which will give me 8x. And 4 times 10-- or 4 times negative 10 because of the minus-- will give me negative 40. And that's still equal to 20.

So now that I'm simplified, I can use my order of operations backwards. So starting by canceling out any adding or subtracting, I'm going to add 40 to both sides to cancel out the subtracting 40. This will leave me with 8x is equal to 60.

Now I can cancel out the multiplying by 8 by dividing both sides by 8. On this side, my 8's will cancel, and I've isolated my x variable. And here I can simplify this 60 divided by 8 is 7.5 for my final answer.

Now there is another way that you could solve this equation. Instead of distributing the 4 times both numbers in the parentheses, you can simply cancel out the 4 that's being multiplied. So if I were to divide both sides by 4, then my equation would just become 2x minus 10 is equal to 5. And then you could solve it as we did the other example. So either method will work.

Here's my last example. If you're feeling pretty confident with the examples we've done, go ahead and pause and try it on your own and then check back. Just remember that you want to start by simplifying if you can, and then use order of operations backwards to isolate the variable.

So, we can simplify this example because I have two like terms-- negative 6x and a positive 2x. So combining those negative 6 plus 2 is going to give me a negative 4x. Then I can bring down my minus 4 and bring down my equals 7.

So now I can solve this. I'm going to use order of operations backwards starting by canceling out any adding or subtracting. So to cancel out minus 4, I'm going to add 4 on both sides. These will cancel, and I'll be left with negative 4x is equal to 11.

And now I just need to cancel out my negative 4 that's being multiplied by the x. So I'm going to divide both sides by negative 4. These will cancel, and I'll be left with x is equal to negative 2.75. Again, you can leave your answer as a fraction if you want, but I converted it into a decimal.

So let's go over our key points from today. As usual, make sure you get them in your notes so you can refer to them later.

To solve an equation, isolate the variable you want to solve for. To isolate variables, use inverse operations to cancel the operations around the variable. And in using the inverse operations, use the order of operations in reverse order.

So I hope that these key points and examples helped you understand a little bit more about isolating variables. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.