Linear equations 1

Welcome to level one linear equations. So let's start doing some problems. So let's say I had the equation 5-- a big fat 5, 5x equals 20. So at first this might look a little unfamiliar for you, but if I were to rephrase this, I think you'll realize this is a pretty easy problem. This is the same thing as saying 5 times question mark equals 20. And the reason we do the notation a little bit-- we write the 5 next to the x, because when you write a number right next to a variable, you assume that you're multiplying them. So this is just saying 5 times x, so instead of a question mark, we're writing an x. So 5 times x is equal to 20. Now, most of you all could do that in your head. You could say, well, what number times 5 is equal to 20? Well, it equals 4. But I'll show you a way to do it systematically just in case that 5 was a more complicated number. So let me make my pen a little thinner, OK. So rewriting it, if I had 5x equals 20, we could do two things and they're essentially the same thing. We could say we just divide both sides of this equation by 5, in which case, the left hand side, those two 5's will cancel out, we'll get x. And the right hand side, 20 divided by 5 is 4, and we would have solved it. Another way to do it, and this is actually the exact same way, we're just phrasing it a little different. If you said 5x equals 20, instead of dividing by 5, we could multiply by 1/5. And if you look at that, you can realize that multiplying by 1/5 is the same thing as dividing by 5, if you know the difference between dividing and multiplying fractions. And then that gets the same thing, 1/5 times 5 is 1, so you're just left with an x equals 4. I tend to focus a little bit more on this because when we start having fractions instead of a 5, it's easier just to think about multiplying by the reciprocal. Actually, let's do one of those right now. So let's say I had negative 3/4 times x equals 10/13. Now, this is a harder problem. I can't do this one in my head. We're saying negative 3/4 times some number x is equal to 10/13. If someone came up to you on the street and asked you that, I think you'd be like me, and you'd be pretty stumped. But let's work it out algebraically. Well, we do the same thing. We multiply both sides by the coefficient on x. So the coefficient, all that is, all that fancy word means, is the number that's being multiplied by x. So what's the reciprocal of minus 3/4. Well, it's minus 4/3 times, and dot is another way to use times, and you're probably wondering why in algebra, there are all these other conventions for doing times as opposed to just the traditional multiplication sign. And the main reason is, I think, just a regular multiplication sign gets confused with the variable x, so they thought of either using a dot if you're multiplying two constants, or just writing it next to a variable to imply you're multiplying a variable. So if we multiply the left hand side by negative 4/3, we also have to do the same thing to the right hand side, minus 4/3. The left hand side, the minus 4/3 and the 3/4, they cancel out. You could work it out on your own to see that they do. They equal 1, so we're just left with x is equal to 10 times minus 4 is minus 40, 13 times 3, well, that's equal to 39. So we get x is equal to minus 40/39. And I like to leave my fractions improper because it's easier to deal with them. But you could also view that-- that's minus-- if you wanted to write it as a mixed number, that's minus 1 and 1/39. I tend to keep it like this. Let's check to make sure that's right. The cool thing about algebra is you can always get your answer and put it back into the original equation to make sure you are right. So the original equation was minus 3/4 times x, and here we'll substitute the x back into the equation. Wherever we saw x, we'll now put our answer. So it's minus 40/39, and our original equation said that equals 10/13. Well, and once again, when I just write the 3/4 right next to the parentheses like that, that's just another way of writing times. So minus 3 times minus 40, it is minus 100-- Actually, we could do something a little bit simpler. This 4 becomes a 1 and this becomes a 10. If you remember when you're multiplying fractions, you can simplify it like that. So it actually becomes minus-- actually, plus 30, because we have a minus times a minus and 3 times 10, over, the 4 is now 1, so all we have left is 39. And 30/39, if we divide the top and the bottom by 3, we get 10 over 13, which is the same thing as what the equation said we would get, so we know that we've got the right answer. Let's do one more problem. Minus 5/6x is equal to 7/8. And if you want to try this problem yourself, now's a good time to pause, and I'm going to start doing the problem right now. So same thing. What's the reciprocal of minus 5/6? Well, it's minus 6/5. We multiply that. If you do that on the left hand side, we have to do it on the right hand side as well. Minus 6/5. The left hand side, the minus 6/5 and the minus 5/6 cancel out. We're just left with x. And the right hand side, we have, well, we can divide both the 6 and the 8 by 2, so this 6 becomes negative 3. This becomes 4. 7 times negative 3 is minus 21/20. And assuming I haven't made any careless mistakes, that should be right. Actually, let's just check that real quick. Minus 5/6 times minus 21/20. Well, that equals 5, make that into 1. Turn this into a 4. Make this into a 2. Make this into a 7. Negative times negative is positive. So you have 7. 2 times 4 is 8. And that's what we said we would get. So we got it right. I think you're ready at this point to try some level one equations. Have fun.