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# Two-step equations with decimals and fractions

CCSS.Math:

## Video transcript

let's do a few more examples solving equations and I think you're going to see that these equations require require a few more steps than the ones we did in the last video but the fun thing about these is that there's more than way one way to do it but as long as you do legitimate steps as long as anything you do to the left-hand side you also do to the right-hand side you should move in the correct direction or you shouldn't get the wrong answer so let's let's do a couple of these so the first one says one two I'll rewrite it one point three times X minus 0.7 times X is equal to 12 well here the the first thing that my instinct is to do is to to merge these two terms because I have one point three of something minus 0.7 of that same something this is the same variable if I have one point three apples minus 0.7 apples well why don't I subtract 0.7 from one point three and I will get so I will get one point three minus zero point seven if there's X's or apples or whatever you want to call them so is equal to twelve you can imagine that I did the reverse distributive property out here I factored out an X by the way my head thinks about it is I have one point three of something minus point seven of something that's going to be equal to one point three minus point seven of those some things that X and of course one point three minus 0.7 is 0.6 0.6 times X of my something's is equal to 12 and now this looks just like one of the problems we did in the last video we have a coefficient times X is equal to some other number well let's divide both sides of this equation by that coefficient divide both sides by 0.6 divided both sides by 0.6 and what is so the left-hand side will just become an X X is equal to and what is 12 divided by 0.6 0.6 goes into 12 let's add some decimal points here that's the same thing as 6 going into 120 6 goes into 12 2 times 2 times 6 is 12 subtract you get a Z row 6 goes into 0 zero times it's going to go 20 times 12 divided by 0.6 is 20 and we can verify what's let's verify this 1 point 3 let's substitute back 1.3 times 20 minus 0.7 times 20 let's verify that that is equal to 12 so I'll take the calculator out just so you don't have to trust my math so we have 1.3 times 20 is equal to 26 so this piece right here that piece right there is 26 and then 0.7 times 20 that's I don't need a calculator for that that is 14 26 minus 14 is 12 so it checks out we got the right answer for this equation X is equal to 20 let's do this one right here 5 X minus 3 X plus 2 is equal 1 this looks very complicated and whenever something looks daunting just do steps that look like they're simplifying the equation and over time as long as you do the legitimate steps you should be able to make some progress so the first thing I want to do is I want to distribute this negative one over here so this is the same thing as 5 X minus 3 X minus 3 X minus 2 alright I just did the distributive property on the 3 X which is a negative 1 times 3 X plus 2 so it's negative 1 times 3 X plus negative 1 times 2 or negative 3 X minus 2 and that is going to be equal to 1 now I have five of something minus 3 of that same something so that's going to be equal to 2 of that something 5x minus 3x is 2x 5 minus 3 is 2 and then I have the minus 2 is equal to 1 and now I like to get it to the form where I have 2 X or I have something times X is equal to something so I want to get rid of this negative 2 on the left hand side the best way I know how to do that is to add 2 to both sides so add it to on the left hand side if I do it to the left hand side I've got to do it to the right hand side plus 2 on the right hand side the these two guys will cancel out and you're going to get 2x you're going to get 2x is equal to one plus two is equal to three and now you can divide both sides by 2 and you get X is equal to X is equal to three halves and I'll leave it for you to verify that this is indeed the correct answer let me draw a little line here so that our work doesn't get messy although Matt might have made it even Messier so here we have to solve for s and let me have a fraction and 2's terms how do we do that well just do it the same way we have 1 times s - you could view this as 3/8 times s is equal to 5/6 you could view this as 1 times s minus 3/8 times s is equal to 5/6 you could factor out an S if you like maybe I'll do it this way I'll factor it onto the left hand side this is the same thing as s times 1 minus 3/8 is equal to 5/6 and 1 minus 3/8 what is that that's 1 I can rewrite as 8 over 8 right that's one so this is the same thing as 8/8 minus 3/8 is 5/8 5/8 times s you can switch the order of multiplication 5/8 times s is equal to 5/6 and you might be able to go straight from that if I have 1 of something minus 3/8 of that something I have 8/8 of that something minus 3/8 of that something I'm going to have 5/8 of that something and now to solve for s how I can multiply both sides by the inverse of this coefficient so I multiply 8 over 5 times 5 8 s if I do it to the left hand side I have to do it to the right hand side 8 over 5 I multiplied by 8 over 5 so that those cancel out and those cancel out and you are left with s is equal to right this was just a 1 is equal to well the fives we can divide divide the numerator denominator by 5 divide the numerator by 2 and the denominator by 2 you're left with sorry divide the denominator by 2 you get 6 divided by 2 is 3 you're left four-thirds s is equal to four-thirds let's do one more of these so here I have five times Q minus seven over twelve is equal to two thirds let me write this and like I could rewrite this it's just five over twelve times Q minus seven is equal to two thirds and what I want to do with this video is to show you that I can do it two different ways but as long as I do legitimate operations I should get the same answer so that what the first way I'm going to do it is I'm going to multiply both sides of this equation by the inverse of five twelve so I'm going to multiply both sides by 12 over 5 I'm going to multiply both sides by 12 over five because I want to get rid of this 512 s on the left-hand side it makes everything look a little bit messy and I multiply it by 12 over five because these are going to cancel out the 5 and the 5 cancel out the 12 and the twelve cancel out so the left-hand side of my equation becomes Q minus seven is equal to the right-hand side 2/3 times 512 so if you divide the 12 by 3 you get a 4 you divide the 3 by 3 you get a 1 so 2 times 4 is 8 over 5 and now we can add 7 to both sides of this equation so at let's add I want to do that in a different color add 7 to both sides of this equation these two 7s cancel out that was the whole point of adding the 7 and you are left with Q is equal to 8/5 plus 7 or we could write 8 fifths plus 7 can be written as 35 35 fifths and so this is going to be equal to 8 well the denominator is 5 8 plus 35 8 plus 35 is 43 so my answer going this way is Q is equal to 43 over 5 now I said I would do it two ways let's do it another way so let me write the same problem down so I have 512 512 actually let me do it a completely different way let me say let me write it the way they wrote it five times q minus 7 over 12 is equal to 2/3 let me just get rid of the 12 first let me multiply both sides of this equation by 12 I just don't like that 12 sitting there so I'm going to multiply both sides by 12 so these are going to cancel out and you're going to be left with 5 times Q minus 7 is equal to 2/3 times 12 that's the same thing as 24 over 3 so this is let me write this 2 over 3 times 12 over 1 is equal to what you get a if you divide that by 3 you get a 4 divide that by 3 you get a 1 it's equal to 8 so we get 5 times Q minus 7 is equal to 8 and then instead of dividing both sides by 5 which would get us something pretty close to what we were doing over here let me distribute this 5 I just want to show you you can do it multiple legitimate ways so 5 times Q is 5q 5 times negative 7 is minus or negative 35 is equal to 8/5 Q minus 35 is equal to 8 now if I want to get rid of that minus 35 or that negative 35 the best way to do it is to add 35 to both sides so 35 to both sides I did that so these cancel out and I'm left with 5 Q is equal to 8 plus 35 which is 43 now I can multiply both sides of this equation by 1/5 which is the same thing as dividing both sides by 5 and these cancel out you get Q is equal to 43 over 5 so there's a bunch of ways you can do these problems but as long as you do legitimate steps you will get the right answer and I'll leave it to you to verify that this truly is the right answer for Q this is the Q that will satisfy this equation let's do one word problem here Jade is stranded downtown with only \$10 to get home taxis cost 75 cents per mile but there is an additional 235 higher charge write a formula and use it to calculate how many miles she can travel with her money all right so the total the total cost of a cab ride is going to be equal to just the initial higher charge the initial higher charge which is two dollars and thirty five plus the seventy five cents per mile plus the seventy five cents times the number of miles we're letting M is equal to the miles she travels miles traveled miles traveled so this is the equation we know that she can only she only has ten dollars to get home so her cost has to be ten dollars so we have to say the cost has to be \$10 so 10 is equal to two point three five plus plus 0.75 M so how do we solve for M or the number of miles Jade can travel we can get rid of the two point three five on this right hand side by subtracting that amount from both sides of this equation so let's do that so let's subtract a minus two point three five let's subtract a minus two point three five from both sides these will cancel out that was the point the left-hand side what is 10 minus two point three and I would have three four this is ten minus ten minus two point three five now these will cancel out now what is 10 minus two point three five 10 minus two is 10 minus 2 is 8 10 minus two point three is seven point seven so it's going to be seven point six five if you want to believe me let's do it 10 minus two point three five seven point six five and that is going to be equal to 0.75 and let me write that in that same color it's nice to see where different things came from 0.75 and I have like five shades of this purple here so this is this is that that is that and then these two guys cancelled out now to solve for M I can just divide both sides by 0.75 so I divide that side by 0.75 I have to do it to the left hand side as well 0.75 that cancels out so on the right hand side I'm left with just an M and on the left hand side I'll have to get my calculator out for this one I have seven point six five divided by 0.75 which is equal to ten point two so Jade can travel ten point M is ten point two so Jade can travel ten point two miles