Algebra (all content)
solving equations of the form AX+B=C. Created by Sal Khan.
Welcome to level two linear equations. Let's do a problem. 2x plus 3 is equal to minus 15. Throw the minus in there to make it a little bit tougher. So the first thing we want to do whenever we do any linear equation, is we want to get all of the variable terms on one hand side of the equation and all the constant terms on the other side. And it doesn't really matter, although I tend to get my variables on the left hand side of the equation. Well, my variables are already on the left hand side of the equation but I have this plus 3 that I somehow want to move to the right hand side of the equation. And the way I can-- you can put it in quotes, move the 3 is I can subtract 3 from both sides of this equation. And look at that carefully as to why you think that works. Because if I subtract 3 from the left hand side, clearly this negative 3 that I'm subtracting and the original 3 will cancel out and become 0. and as long as I do whatever I do on the left hand side, as long as I do it on the right hand side as well, because whatever you do on one side of the equal side, you have to do to the other side, then I'm making a valid operation. So this will simplify to 2x, because the 3's cancel out. They become just 0. Equals minus 15 minus 3. Well, that's minus 18. And now, we're just at a level one problem, and you can just multiply both sides of this equation times the reciprocal on the coefficient of 2x. I mean, some people would just say that we're dividing by 2, which is essentially what we're doing. I like to always go with the reciprocal, because if this 2 was a fraction, it's easier to think about it that way. But either way, you either multiply by the reciprocal, or divide by the number. It's the same thing. So 1/2 times 2x. Well, that's just 1x. So you get x equals, and then minus 18/2. And minus 18/2, well, that just equals minus 9. Let's do another problem. And actually, well, if we wanted to check it, we could say, well, the original problem was 2x plus 3 equals minus 15. So we could say 2 times minus 9 plus 3. 2 times minus 9 is minus 18 plus 3. Well, that's equal to minus 15, which is equal to what the original equation said, so we know that's right. That's the neat thing about algebra. You can always check your work. Let's do another problem. I'm going to put some fractions in this time, just to show you that it can get a little bit hairy. So let's say I had minus 1/2x plus 3/4 is equal to 5/6. So we'll do the same thing. First, we just want to get this 3/4 out of the left hand side of the equation, and actually, if you want to try working this out yourself, you might want to pause the video and then play it once you're ready to see how I do it. Anyway, let me move forward assuming you haven't paused it. If we want to get rid of this 3/4, all we do is we subtract 3/4 from both sides of this equation. Minus 3/4. Well, the left hand side, the two 3/4 will just cancel. We get minus 1/2x equals, and then on the right hand side, we just have to do this fraction addition or fraction subtraction. So the least common multiple of 6 and 4 is 12. So this becomes 5/6 6 is 10/12 minus 3/4 is 9/12, so we get minus 1/2x is equal to 1/12. Hopefully, I didn't make a mistake over here. And if that step confused you, I went a little fast, you might just want to review the adding and subtraction of fractions. So going back to where we were. So now all we have to do is, well, the coefficient on the x term is minus 1/2, and this is now a level one problem. So to solve for x, we just multiply both sides by the reciprocal of this minus 1/2x, and that's minus 2/1 times minus 1/2x on that side, and then that's times minus 2/1. The left hand side, and you're used to this by now, simplifies to x. The right hand side becomes minus 2/12, and we could simplify that further to minus 1/6. Well, let's check that just to make sure we got it right. So let's try to remember that minus 1/6. So the original problem was minus 1/2x, so here we can substitute the minus 1/6, plus 3/4. I just wrote only the left hand side of the original problem. So minus 1/2 times minus 1/6, well, that's positive 1/12 plus 3/4. Well, that's the same thing as 12, the 1 stays the same, plus 9. 1 plus 9 is 10 over 12. And that is equal to 5/6, which is what our original problem was. Our original problem was this. This stuff I wrote later. So it's 5/6, so the problem checks out. So hopefully, you're now ready to try some level two problems on your own. I might add some other example problems. But the only extra step here relative to level one problems is you'll have this constant term that you need to add or subtract from both sides of this equation, and you'll essentially turn it into a level one problem. Have fun.