welcome to level one linear equations let's start doing some problems so let's say i had the equation 5 (that's a big fat 5). 5x equals 20 so at first this might look a little unfamiliar to you but if I rephrase this, you'll realize this is pretty easy problem all this is is, it's the same thing as saying this is 5 times questionmark equals 20 and the reason why we 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 saying 5 times x so instead of a questionmark we're writing an x so five times x is equal to 20 now most of you can do that all in your head you can say well what number times 5 is equal to 20? well, it equals 4 but we can do it systematically just in case that 5 is a more complicated number so let me make my pen a little thinner. ok. so rewriting it if I had 5 x equals 20 we could do 2 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 fives will cancel out we'll get x and then the right hand side 20 divided by 5 is 4. And we've solved it. Another way to do it and this is actually the exact same way we are just phrasing it a little different if you said 5 x equals 20 instead of dividing by 5 we could multiply by one fifth and if you look at that you could realize that multiplying by one fifth is the same thing as dividing by five if you know the difference between dividing and multiplying fractions. and then that gets the same thing one fifth times 5 is 1 so you're 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 five it's easier to just think about multiplying by the reciprocle actually let's do one of those right now so let's say I had negative 3 over 4 times x equals 10 over 13 now this is a harder problem I can't do this one in my head We're saying negative three fourths times some number x is equal to 10 over 13 if someone came to you on the street and asked you that I think you'd be like me you'd be pretty .. ah... stumped. but let's work it out algebraically well, 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 the number that's being multiplied by x so what's the coefficient, the reciprocal of minus three fourths? well, it's minus 4 over 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 a traditional multiplication sign and the main reason I think it's just a regular multiplication sign just 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 four thirds we also have to do the same thing to the right hand side minus four thirds the left hand side the minus four thirds and three fourths, they cancel out you could work it out on your own to see that they do they equal one so we're just left with x x is equal to 10 times minus 4 is minus 40 13 times 3 well that's equal to 39 so you get x is equal to minus 40 over 39 and I like to leave my fractions improper cause it's easier to deal with them but you could also... if you wanted to write that as a mixed number minus 1 and one thirthyninth 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 into the orgiginal equation to make sure you're right so the original equation was minus three fourths times x and here we'll substitute the x back into the equation where we saw x we'll now put our answer so it's minus 40 over 39 and our original equation said that that equals to 10 over 13 well. and once again when I just write the three fourths times right next to the parenthesis like that that's another way of writing times so minus 3 times minus 40 is minus 100 ... actually we could do something a little simpler this 4 becomes a 1 and this becomes a 10 and if you remember how when you're multiplying fractions you can simplify it like that so it actually becomes plus 30 (because we have a minus times a minus is plus) and 3 times 10 over the 4 is now a 1 so all is left is 39 and 30 over 39 if we divide the top and the bottom by 3 we get 10 over 13 which is the same thing which the equation said it would be so we know that we got the right answer let's do one more problem minus 5 sixths x is equal to 7 eights and if you wanna try this problem yourself now is a good time to pause And I'm gonna start doing the problem right now So, same thing What's the reciprocal of minus 5 sixths? Well, it's minus 6 over 5 We multiply that If we do it on the left hand side we have to do it on the right hand side as well minus 6 over 5 the left hand side the minus 6 over 5 and minus 5 sixths cancel out we're just left with x and the right hand side we have well we can divide both 6 and the 8 by 2 so that makes negative 3 and this becomes 4 7 times negative 3 is minus 21 over 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 sixths times minus 21 over 20 well that equals we could make this 5 a 1 and turn this into a 4 make this into a 2 make this into a 7 negative times negative is positive so 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 1 equations Have fun!