Welcome to level1 Linear equations, so let’s start doing some problems. So let’s say I had the equation 5, big fat 5, 5X equals 20, so at first this might look a little unfamiliar for you but if I would rephrase this I think you’ll realize that this is a pretty easy problem. All this is, this is the same thing as saying 5 times question mark equals 20.
And the reason why we did the notational, 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 are writing it X, so 5 times X is equal to 20.
Now most of you all could do that in your head you could say 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 this a little, my pen a little thinner okay. So rewriting it if I had 5 X 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 5s will cancel out we’ll get X, and at the right hand side 20 divided by 5 is 4 and we solved it.
Another way to do it and this is actually the exact same way we’re just phrasing a little different if you set 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 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 to just think about multiplying by the reciprocal, actually let’s do one of those right now.
So let’s say I had negative ¾ times X equals to, equals 10/13. Now, this is a harder problem I can’t do this one in my head we’re saying negative ¾ times some number X is equal to 10/13. If someone came up to you on the street and asked you that I, I think you’d be like me and you’d be pretty stumped but let’s work it out algebraically.
But we’ll 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 a number that’s being multiplied by X. So what's the reciprocal of minus ¾? Well, it’s minus 4/3 times and dot is another way to use times and you probably wondering why algebra—there 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 your multiplying 2 constants or just writing it next to variable to imply your multiplying a variable.
So if you multiply the left hand side by negative 4/3, we also 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 and see what 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 live my fractions improper because it’s easier to deal with them but you could just you could also view that if that’s minus if you want to write as a mixed number, minus 1 and 130. I tend to keep it like this though. 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 to 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 where we saw X when I’ll put our answer. So it's minus 40 over 39 and our original equation said that’s equal is 10/30.
Well, and once again, when I just write the 3/4 times right next to the parenthesis like that, that’s just another way of writing times. So minus 3 times minus 40 is minus 100, and actually we can do something a little bit simpler. This 4 becomes a 1 and this becomes a 10. If you remember how, when your multiplying fractions you can simplify it like that. So it actually becomes minus -- actually plus 30, because we have a minus times a minus 3 times 10 over the 4 is now 1, so we have just left 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 got the right answer.
Lets do one more problem, minus 5/6X is equal to 7/8. And if you want to try this problem yourself, now is the good time to pause and I’m gonna 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 it on the left hand side, we have to do 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 just left with X and the right hand side, we have -- well we can divide with the 6 and the 8 by 2, so the sums 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 fast. So minus 5/6 times minus 21/20, well that equals what is 5 we could make that a 1 turn this into a 4, and make this into a 2, make this into a 7 negative times negative is positive, so if 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!