Linear equations with variables on both sides
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We have the equation 20 minus 7 times x is equal to 6 times x minus 6. And we need to solve for x. So the way I like to do these is we just like to separate the constant terms, which are the 20 and the negative 6 on one side of the equation. I'll put them on the right-hand side. And then we'll put all the x terms, the negative 7x and the 6x, we'll put it all on the left-hand side. So to get the 20 out of the way from the left-hand side, let's subtract it. Let's subtract it from the left-hand side. But this is an equation, anything you do to the left-hand side, you also have to do to the right-hand side. If that is equal to that, in order for them to still be equal, anything I do to the left-hand side I have to do to the right-hand side. So I subtracted 20 from the left, let me also subtract 20 from the right. And so the left-hand side of the equation, 20 minus 20 is just 0. That was the whole point, they cancel out. Don't have to write it down. And then I have a negative 7x, it just gets carried down. And then that is equal to the right-hand side of the equation. I have a 6x. I'm not adding or subtracting anything to that. But then I have a negative 6 minus 20. So if I'm already 6 below 0 on the number line, and I go another 20 below that, that's at negative 26. Now, the next thing we want to do is we want to get all the x terms on the left-hand side. So we don't want this 6x here, so maybe we subtract 6x from both sides. So let's subtract 6x from the right, subtract 6x from the left, and what do we get? The left-hand side, negative 7x minus 6x, that's negative 13x. Right? Negative 7 of something minus another 6 of that something is going to be negative 13 of that something. And that is going to be equal to 6x minus 6x. That cancels out. That was the whole point by subtracting negative 6x. And then we have just a negative 26, or minus 26, depending on how you want to view it, so negative 13x is equal to negative 26. Now, our whole goal, just to remember, is to isolate the x. We have a negative 13 times the x here. So the best way to isolate it is if we have something times x, if we divide by that something, we'll isolate the x. So let's divide by negative 13. Now, you know by now, anything you do to the left-hand side of an equation, you have to do to the right-hand side. So we're going to have to divide both sides of the equation by negative 13. Now, what does the left-hand side become? Negative 13 times x divided by negative 13, that's just going to be x. You multiply something times x, divide it by the something, you're just going to be left with an x. So the left-hand side just becomes an x. x is equal to negative 26 divided by negative 13. Well, that's just positive 2, right? A negative divided by a negative is a positive. 26 divided by 13 is 2. And that is our answer. That is our answer. Now let's verify that it really works. That's the fun thing about algebra. You can always make sure that you got the right answer. So let's substitute it back into the original equation. So we have 20 minus 7 times x-- x is 2-- minus 7 times 2 is equal to 6 times x-- we've solved for x, it is 2-- minus 6. So let's verify that this left-hand side really does equal this right-hand side. So the left-hand side simplifies to 20 minus 7 times 2, which is 14. 20 minus 14 is 6. That's what the left-hand side simplifies to. The right-hand side, we have 6 times 2, which is 12 minus 6. 12 minus 6 is 6. So they are, indeed, equal, and we did, indeed, get the right answer.