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let's try some slightly more complicated equations let's say we have three times X three times X plus five plus five I want to make sure I get all the colors nice plus five is equal to is equal to is equal to seventeen is equal to seventeen so what's different about this and what we saw in the last video is all of a sudden now we have this plus five if it was just three X is equal to seventeen you could divide both sides by three and you would get your answer but now this five seems to mess things up a little bit now before we even solve it let's think about what it's saying let's solve it kind of in a tangible way and then we'll solve it using operations that hopefully will make sense after that so 3 times X literally means so let me write it over here so we have 3 times X so you literally have an X plus an X plus an X that right there is a 3x and then that's plus 5 and I'm actually going to write it out as five objects so plus one two three four five that this this right here is three X plus five is equal to 17 is equal to 17 so let me write the equal sign is equal to 17 now let me draw 17 objects here so 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 now these two things are equal so anything you do to this side you'd have to do that side if we were to get rid of one object here you'd want to get rid of one object there in order for the Equality to still be true now what can we do to both sides of this equation so we can get it in the form that we're used to where we only have a 3x on the left hand side where we don't have this 5 well ideally we would just get rid of these five objects here you would literally get rid of these five objects 1 2 3 4 5 but like I said if the original thing was equal to the original thing on the right if we get rid of 5 objects from the left hand we have to get rid of five objects from the right hand side so we have to do it here too one two three four five now what is a symbolic way of representing taking away five things well you're subtracting five from both sides of this equation so that's what we're doing here when we took away five from the left and from the right so we're subtracting five from the left that's what we did here and we're also subtracting five from the right do that right over there now what does the left-hand side of the equation now become the left-hand side you have five minus five these cancel out you're just left with the 3x you are just left with it's a different shade of green you are just left with the three the 3x the five and the negative five canceled out and you see that here when you got rid of these five objects we were just left with the three X's this right here is the 3x and the whole reason why we subtracted five is because we wanted this five to go away now what does the right-hand side of the equation look like so it's three X is going to be let me write the Equality sign right under it is equal to well you can either just do it mathematically say okay 17 minus five 17 minus five is 12 or you could just count over here I had 17 things I took away five I have 12 left one two three four five six seven eight nine ten eleven twelve twelve that's what subtraction is it's just taking away five things so now we have it in a pretty straightforward form 3x is equal to 12 all we have to do is divide both sides of this equation by three so we have a so we're just left with an X on the left hand side so we divide by let me pick a nicer color than that let me do let me do this pink color so you divide the left-hand side by three the right-hand side by three and remember what that's equivalent to the left-hand side now that this stuff exists anymore so we should ignore it none of this stuff exists anymore in fact let me let me clear it out just so that we don't even have to look at it we subtracted it so let me clear it let me clear it over here let me clear it over here and so now we are dividing both sides by 3 divide the left-hand side by 3 2 1 2 two three so three groups each of them have an X in it if you divide this right-hand side by three let's see you have one two and three so it's three groups of four so when you do it mathematically here the 3s cancel out three times something divided by three is just two something so you're left with X is equal to and then 12 divided by 3 is 4 12 divided by 3 is 4 you get X is equal to 4 you get that exact same thing over here when you divide it 3x into groups of three each of the groups had an X in it and when you divided 12 into groups of 3 each of the groups have a 4 in it so X must be equal to 4 X is equal to 4 let's do another one and this time I won't draw it all out like this but hopefully you'll see that the same type of processes are involved let's say I have let me scroll down a little bit let's say I have 7 X so 7 X 7 X I'll go slightly more complicated one this time 7 X minus 2 is equal to is equal to let's say is equal to I'll make the numbers not work out nice and clean is equal to negative 10 now this all of a sudden becomes a lot more maybe you know you have a negative sign we have a negative over here but we're going to do the exact same thing the first thing we want to do if we want to get the left hand side simplified to just 7x is we want to get rid of this negative 2 and what can we add or subtract to both sides equation to get rid of this negative 2 well if we add 2 to the left hand side these two guys will cancel out but remember this is equal to that if we want the Equality still hold if we add 2 to the left hand side we also have to do it to the right hand side so what is the new left-hand side going to be equal to so we have 7 X 7 X negative 2 plus 2 is just 0 I could write plus a 0 or I could just write nothing there and I'll just nothing so we get 7 X is equal to now what's negative 10 plus 2 and this is a little bit of review of adding and subtracting negative numbers remember I'll draw the number line here for you if I draw the number line so if this is 0 this is 1 we could keep going in the positive direction negative 10 is out here negative 10 negative 9 negative 8 negative 7 there's a bunch of numbers here you know dot dot I don't have space to draw them all but we're starting at negative 10 and we're adding 2 to it so we're moving in the positive direction on the number line so we're going 1 2 so it's negative 8 don't get confused don't say ok 10 plus 2 is 12 so negative 10 plus 2 is negative 12 no negative 10 minus 2 would be negative 12 because you'll be going more negative here we're having a negative number but we're going to the right we're going in the positive direction so this is negative 8 so we have 7 X is equal to negative 8 so now you might be saying well how do i how do I do this type of a problem and you know I have a negative number you do the exact same way we want if we want to just have an X on the left hand side we have to divide the left hand side by 7 so that the 7x divided by 7 just 7s cancel out you're left with X so let's do that if you divide by 7 those cancel out but you can't just do it to the left-hand side anything you do the left you have to do the right in order for the Equality to still hold true so let's divide the right by 7 as well and we are left with just an X just and X is equal to negative 8 divided by 7 we could work it out it'll be some type of a decimal if you were to use a calculator or you could just leave it in fraction form negative 8 divided by 7 is negative 8 sevenths negative 8 7 or if you want to write it as a mixed number X is equal to 7 goes into 8 one time and has a remainder of 1 so it's negative 1 and 1/7 either one would be acceptable