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# Linear equations 1

## Video transcript

welcome to level one linear equations so let's start doing some problems so let's say I had the equation five let's big fat five five x 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 all this is this is the same thing as saying five times question mark equals 20 and the reason why 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 five 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 can 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 this a little my pen little thinner okay 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 fives will cancel out we'll get X and then 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 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 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 so let's say I had negative 3 over 4 times X equals 2 equals 10 over 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 over 13 if someone came up to you on the street and asked you that I think you'd be like me and then 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 cuts the reciprocal of minus 3/4 well it's minus 4 over 3 times and dot is another way to use x and you're probably wondering why in algebra there are all these other conventions for doing x as opposed to just a traditional multiplication sign and the main reason is I think 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 one so we're just left with X is equal to 10 x minus 4 is minus 40 13 times 3 well that's equal to 39 so we get X is equal to minus 40 over 39 and I like to leave my fractions improper because it's easier to to deal with them but you could just you could also view that if that's - if you want to write it as a mixed number - 1 + 139 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 original equation and make sure you were right so the original equation was minus 3/4 times X and here we'll substitute the X back into the equation where we wherever we saw X will now put our answer so it's minus 40 over 39 and our original equation said that that equals 10 over 13 well and once again when I just write the 3/4 times the Perret right next to the parenthesis like that that's just another way of writing x so minus 3 times minus 40 is minus 100 and actually we could we could do something a little bit simpler this 4 becomes a 1 and this becomes a 10 if you remember how 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 once all we have is left just 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 as what the equation said it we would get so we know that we got the right answer let's do one more problem minus 5/6 X is equal to 7/8 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 six over five we multiply that if you do it on the left hand side we have to do it on the right hand side as well minus six over five the left hand side the minus six over five of the minus 5/6 cancel out we're just left with X and the right hand side we have well we can take the set the the we could divide both the 6 and the 8 by 2 so this negative 3 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 5/6 times minus 21 over 20 well that equals on this five we could make that a 1 turn this into a 4 make this into a to make this into a 7 negative times negative is positive save 7 2 times 4 is 8 and that's what we said we would get so we got it right I think you're you're ready at this point to try some level 1 equations have fun