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Math is cool and you can do it.
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Moving on to solving linear equations part 4,
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before doing a few more examples, I want to go over how to solve equations
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of this form where you have negative x equals 5, or negative x equals negative 4.
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Because this is not solved for x. This is what this says, well, the opposite of x is 5.
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So the simple way to do this is you just take the opposite of both sides.
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Take the opposite of both sides, which means you change the signs
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of all terms on each side of the equation.
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Now when you do this, by the way, this is like multiplying both sides by negative 1.
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This actually is another property of equality.
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You can multiply both sides of an equation by the same number.
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We'll be doing that in another video.
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But for now, let's work on this.
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Negative x equals negative 5.
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Negative x equals 5 is equivalent to just writing x equals negative 5.
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That will be the answer.
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If you had negative x equals negative 4, you could just write that.
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That's the same thing as x equals 4, and now you've solved for x.
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Think about it. The opposite of what number is 5?
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Oh, the opposite of negative 5 would be 5. That's why x is negative 5.
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This is saying the opposite of what is negative 4?
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The opposite of 4 is negative 4.
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You can see when you check it how that is going to make sense.
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If you're going to do the checks to each of these negative x equals 5,
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then you're going to plug in negative 5 for x.
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Should be convinced that's true.
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When we check this other one, negative x equals negative 4.
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We're going to plug in 4 for x.
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You get the same number on both sides.
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That's how you deal with it. You have a minus sign in front of the variable.
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Check this out.
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What if you had something like 2x?
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Let's make this negative 2x plus 3 equals negative x minus 9.
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Anytime you have a minus sign in front of a variable,
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if you don't want it to be a minus sign, if you'd rather be a plus sign,
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you could just change the signs of all the terms if you'd like.
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So watch what would happen here.
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If I changed all the signs, that would be taken the opposite of both sides of the equation.
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You could write 2x minus 3 equals x plus 9.
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See how that works?
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Then I have a positive in front of this 2x.
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Sometimes you can't get a positive in front of both of them. That's going to depend on it.
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Let's finish solving this.
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It's simplified on each side.
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Now we could subtract x from both sides so that I have all the x terms on the left side of the equation.
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At the same time, let's get rid of this minus 3 by adding 3 to both sides of the equation.
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I get for my answer 2x minus x, it's just x equals 12.
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Now let's take the original problem, 2x plus 3 equals negative x minus 9
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and not take the opposite of both sides and see what happens.
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Let's say I want to get my variable on the left so I would add x to both sides.
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I have the variables on the same side of the equation.
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Then I have plus 3. I need to get rid of that so I would be subtracting 3 from both sides.
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Now here's where the trick comes in.
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You have negative 2x plus 1x because x is the same as 1x.
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That will give you a negative x or a negative 1x.
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Here's the problem.
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Negative x equals negative 12 because you have a negative 9 plus negative 3 over here.
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Here's the situation with what do you do?
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You have a minus sign in front of it, now take the opposite of both sides and you would still get x equals 12.
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At some point you would be taking the opposite of both sides.
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Unless you put the variables on the right hand side.
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The thing about algebra, there are so many ways to solve this exact same problem.
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Let's do it one more way.
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The original problem was negative 2x plus 3 equals negative x minus 9.
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You could have added 2x to both sides.
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So the variables on the right hand side and you'll end up with a positive x.
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That would have also worked.
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Then you would have to add 9 to both sides so that the constants, the constants, remember those are the numbers?
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It would be on the left hand side.
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So on the left hand side I have x equals 12 but remember x equals, I mean 12 equals x is the same thing as x equals 12.
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So that's how three different students might do this.
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Now don't get confused, you don't have to do it three ways.
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You just have to do it one way.
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Whatever makes the most sense to you, do it that way.
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Now let's check and make sure x equals 12 is the correct answer.
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I imagine it is because we did it three ways and we got the same answer all three ways.
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So we're going to check that x equals 12 is the correct answer to this problem.
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Remember we write the original problem down, underline it, do the little bar thing and we're going to plug in 12 for x into order of operations on both sides.
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So on the left hand side when I put in 12, doing order of operations, I multiply negative 24 plus 3.
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And negative 24 plus 3 is negative 21.
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And now we're going to do the right hand side plugging in 12 for x. So that's minus, I could just write the 12 if you want you could put parentheses around the 12, but unnecessary.
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And remember that means negative 12 plus negative 9, so that's negative 21.
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So the answer to this problem, yes, just like we suspected, it was 12 or you could write it using the braces.
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Those are called solution brackets or solution braces. There's 12.
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All right, let's try this problem. 3 minus x equals 7.
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This doesn't look bad compared to what we just did, but do notice there's a minus sign in front of the x and that's where we have to be careful.
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So I've got really the extra moly on one side.
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So let's just leave it there and subtract 3 from both sides.
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Here's you have to be careful. A lot of people will write x equals 4.
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Back on, I'm going to do it wrong. Is that how, would that be correct? x equals 4. So let's just check it.
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3 minus x equals 7.
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But we put in 4. Okay, some people like to trick me and they say the answer to that 7, but of course 3 minus 4 is not 7.
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No, no, no, no. Really do it. What's 3 minus 4? That's 3 plus negative 4, which is negative 1.
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Ah, so it does not check. Of course it doesn't check. I hope you all notice that because when you bring this down, there's a minus sign in front of that x.
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That's why I didn't check. So hopefully you would go back and say, ah, there's my mistake.
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And what do you do if you have a minus x equals 4 or the opposite of x? You just change the signs of both sides.
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The answer is x equals negative 4. Now let's check.
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3 minus x equals 7. You're going to plug in negative 4 for x. So 3 minus negative 4 is the same thing as 3 plus 4, which is 7.
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And that's how you know that's the correct answer. So the correct answer is negative 4 or you could use braces.
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If you're going to put a box around your answer, I like to put a box around my answer. It's like you can find it on the page.
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Somebody might leave a blank for you to put your answer. If you're taking a test, that works as well.
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OK, here's one bear of a problem. Put it on pause. Get out your own piece of paper and take your time doing this problem and then checking and seeing if you got the correct answer.
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OK, here I'm going to do it fairly quickly. We need to simplify each side. So I need to do the distributive property here, 2x plus 4 minus 8.
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On the right side, also do the distributive property, 8x minus 12. And when I do the distributive property on the second part, remember you're distributing a negative 5.
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So that's going to be negative 5x and minus 5. You still have to combine like terms. So we have 2x. 4 minus 8 is a negative 4.
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And on the right side, 8x minus 5x is 3x. Minus 12 and minus 5. That's adding a negative 12 and negative 5 is negative 17.
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All right, now I need to get the variables on the left and constants on the right or vice versa. So I'm going to go ahead and subtract 3x from both sides so they're on the left-hand side.
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And since the variables are going to be on the left, I'm going to go ahead and add 4 to both sides.
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So the constants are on the right. And look what happens here. 2 minus 3, I'm going to have that negative x is equal to negative 13.
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And how do we fix that? Take the opposite of both sides, so x equals positive 13. So the answer is positive 13.
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At this point, you could have put the variables on the right. Actually, when I have something in this form, I look and see which is the bigger number, the 3 is bigger than 2.
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So you could have subtracted 2x from both sides, making the variables on the right, and then add 17 to both sides. And you would have gotten x equals 13 without the minus sign in front of it.
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So your answer will be 13. I'm going to leave the check to you. You would go back up to the original problem, put in 13 for x, do order of operations completely on each side, and see if you get the same number on both sides.
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Hey, keep practicing, have fun, it's the toughy. I hope you learned something, remember math is cool.