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# Two-step equations with decimals and fractions

CCSS.Math:

## Video transcript

let's get some practice solving equations that involve fractions and decimals and this equation clearly involves fractions let's see we have negative one-third is equal to J over 4 minus 10 over 3 so encourage you to pause the video and see if you could solve for J what J would make this equation true alright now let's work through this together so what I like to do is I like to isolate the variable that I'm trying to solve for on one side and since it's already on the right hand side let's try to get all the things that involve J on the right hand side and then get rid of everything else on the right hand side so I want to get rid of this negative 10 thirds and the best way I can think of doing that is by adding 10 thirds now I can't just do that to one side of the equation then it wouldn't be equal anymore if this is equal to that in order for the Equality to be true whatever I do to this I have to do to that as well stuff to add ten thirds to both sides I have to add ten thirds to both sides of the equation and so what am I going to get on the on the left hand side I'm gonna have negative one-third plus ten thirds which is 9/3 nine thirds and then that's going to be equal to and on the right hand side the negative ten thirds and the positive ten thirds those cancel out to just zero and I'm just left with J over four it's equal to J over four now you might recognize 9 over 3 that's the same thing as 9 divided by 3 so this is just going to be three so that simplifies it a little bit 3 let me just rewrite it so you don't get confused 3 is equal to J over 4 now to solve for J I could just multiply both sides by 4 because if I divide something by 4 and then multiply by 4 armours gonna be left with that something if I start with J and I divide by 4 and then I multiply and then I multiply by 4 so I'm just gonna multiply it by 4 then I'm just gonna be left with J on the right hand side but I can't just multiply the right-hand side by 4 I have to do it to the left hand side as well so I'll multiply the left-hand side by 4 as well and what I will be left with 4 times 3 is 12 and then J divided by 4 times 4 well that's just going to be J so we get J is equal to 12 and the neat thing about equations is you can verify that you indeed got the right answer you couldn't you can substitute forge a here and verify that negative one-third is equal to 12 over 4 minus 10 thirds does this actually work out well 12 over 4 is the same thing as 3 and if I wanted to write that as thirds this is the same thing as 9 thirds and 9 thirds minus 10 thirds is indeed equal to negative 1/3 so we feel very good about that let's do another example so I have n over 5 plus 0.6 is equal to 2 so let's isolate the end this term that involves an on the left hand side so let's get rid of this 0.6 so let's subtract 0.6 from the left hand side but I can't just do it from the left but to do it from both sides if I want the Equality to hold true so subtract 0.6 now on the left hand side I'm just going to be left with and over 5 and on the right hand side 2 minus 0.6 that's going to be 1.4 and if you won't if you don't want to do this in your head you could work this out separately it's going to be 2 point 0 minus 0.6 you could say oh this is 20 tenths minus 6 tenths which is going to be 14 tenths which is that there or if you want to do it a little bit kind of the traditional method you say oh I'm trying to subtract 6 from 0 let me regroup that's gonna be a 10 I'm gonna take from the ones place 1 if I take a 1 like from the ones place and that's going to be equal to 10 tenths 10 tenths minus 6 tenths is 4/10 and then bring down 1 1 minus 0 ones is just 1 so it's 1 point 4 and now to solve for n row on the Left I've n being divided by 5 I just want any way I could just multiply by 5 so if I multiply by 5 5 times then divided by 5 is gonna be just n but I can't just multiply the left-hand side by 5 to multiply the right-hand side by 5 as well and so what is that going to get us we are going to get n is equal to one point four times five one point four times five now you might be able to do this in your head because this is one in two fifths so this thing should all be cool to seven but I'll just do it this way as well five times four is twenty regroup the to one times five is five plus two is seven and when I look at all the numbers that I'm multiplying I have one digit to the right of the decimal point so my answer will have one digit to the right of the decimal point so it's 7.0 or just seven and is equal to a seven and you can verify this works because seven divided by five is going to be equal to one point four plus zero point six is equal to two let's do one more example this is too much fun all right zero point five times the whole quantity are plus two point seven five is equal to three and there's a bunch of ways that you could tackle this a lot of times when you see something like this your temptation might be let's distribute the zero point five but that makes it a little bit hairy because zero point five times two point seven five you can calculate that and you'll get the right answer if you do it correctly but a simpler thing might be well let's just divide both sides by 0.5 that way I'm going to get more whole numbers involved so if I divide remember whatever I do the left-hand side I have to do the right-hand side and the way my brain thought about it well if I divide by 0.5 on the left hand side I could get rid of this and if I divide by 0.5 on the right hand side I'm still going to get an integer 3 divided by 0.5 is 6 it's the same thing as 3 divided by half how many halves fit into 3 6 halves fit into 3 so this is going to be 6 right over here so these cancel out and then this is going to be equal to 6 so the whole thing is simplified now to our plus two point seven five is going to be equal to is equal to 6 and now to just isolate the are on the left hand side I could subtract the two point seven five from the left but like we've seen multiple times I can't just do it from I can't just do it from the left two points I'm having my brain is malfunctioning two point seven five I can't just do it from the left I have to do it from the right as well two point seven five so this simplifies to R this simplifies to R is equal to what's six minus two point seven five well if you want to do it in your head 6 minus 2 would be 4 and then if you take 0.75 from that it would be 3 point 2 5 if you don't feel comfortable doing in your head we could just write it out six point zero zero minus two point seven five be careful to align the decimals and then I got to do some regrouping let's see I have zero hundreds trying to subtract five hundredths that's not going to work out so I try to regroup from here but I have nothing here so let me regroup from here one one let me take away one of these ones so when I have five ones and then that's gonna be equivalent so have five ones here and that one one I took away is gonna be ten tenths and then I could take one of those tenths away so I'm gonna have nine tenths and that's going to be ten hundredths and now I could subtract ten hundredths minus five hundredths is five hundredths nine tenths minus seven tenths is two tenths my decimals going to be there 5 minus 2 is 3 three point two five and you can verify that three point two five plus two point seven five is six times zero point five is indeed equal to three so we feel once again really good about this