# Rearranging the Combined Gas Equation

here we're going to look at how to rearrange the

gas

### equation

so that you can solve for any of these variables now sometimes this can be a bit of a pain and takes a lot of steps to get to the final point so what I'm going to do here is I'm going to start by doing two examples with a shortcut a trick that make things faster and easier you may be unfamiliar with this trick this might be new to you so I'm going to start with that and then I'm going to do two examples at
the end using the more traditional method of doing a lot of multiplication and division so that we can cancel things back and forth so let's start right out I am going to show you how I'm going to solve for t2 using the shortcut method two things to keep in mind whenever we're

## rearranging

### equation

s like this we need to solve for variable we need to make sure that this variable is all by itself on one side of the

### equation

and the other thing we need to make sure is is that the
variable is on top of the fraction okay so even if we get it by itself but it's still at the bottom that's no good it has to be by itself and on the top in the numerator so anyway t2 solving for this guy here's what we do for the shortcut we cross multiply which means that we do this times this p1 times v1 times t2 that was this equals this multiplied together so t1 times p2 times v2 now this is good because right away it gets t1 and t2 out of the denominator out of the bottom of
fraction so we don't have to worry about that anymore now everything is on top so solving for t2 let's divide both sides by p1 and v1 to get rid of them from this side divided by p1 times v1 and we have to do the same thing to the other side now we have p1 over p1 they cancel out and we have v1 over v1 so they cancel out and we're left with t2 equals t1 times p2 times v2 divided by p1 times v1 let's do one more example with a shortcut method here's two

#### combined

this time let's solve for v2 I'm going to cross multiply again so I'm going to do p1 times v1 times t2 equals the cross multiplication of this equals t1 times p2 times v2 now we want to get v2 by itself so let's divide by p1 and by p2 so we can get rid of those from this side of the

### equation

divided by t1 times p2 divided by t1 times p2 these guys are going to cancel out T 1 over T 1 P 2 P 2 cancels out and we're left with p1 times v1 times t2 divided by t1 times P 2 equals
v2 now you know sometimes people feel uncomfortable when we have the thing that we're solving for on the right hand side right because we're used to saying v2 equals this and so sometimes people don't like it when they said this equals v2 you know what it's totally fine you can flip it if you prefer we can just write v2 equals p1 times v1 times t2 divided by t1 times p2 never worry if you've gotten the variable all by itself on one side but just happens to be the right side
just flip it around if you'd rather have the variable you're solving for B on the left okay so those are the two examples where we use this cross multiplication method for the shortcut if this makes you a little uncomfortable or if maybe your teacher wants you to use a more traditional method now we're going to do two problems where we do a number of steps of multiplication and division so I'm going to start out by solving for P one we want to get P 1 by itself so the first thing
that I'm going to do is get this t1 out of the bottom of the fraction and to do that I'm going to multiply both sides by t1 on top so now we have t1 and you know you can sort of imagine this fraction line goes over here now I have t1 on the top and on the bottom so we get rid of the t1 on this side and we're left with p1 times v1 equals p2 times v2 times t1 divided by t2 okay the next thing that I want to do since I'm solving for P one is to get rid of the v1 that's up here
so I'm going to divide both sides by v1 and now I add v1 to the bottom of the fraction there so have v1 over v1 and those cancel out and I'm left with p1 equals p2 times v2 times t1 divided by t2 times v1 ok one more example here oh and look I made a mistake here this should say this should say t1 there instead of t2 so that's what I want to be solving for I want to be solving for T 1 now if you remember I said earlier that we can never solve some for something if it's in the
bottom of the fraction okay it's the first thing that I need to do is get this guy out of the bottom of the fraction well how can I get it out of the bottom of the fraction I can multiply t1 by both sides both sides by t1 and then we get rid of it from this side but it ends up on the top over here so then we're okay because now it's not on the bottom of the fraction anymore so we rewrite this p2 times v2 times t1 divided by t2 okay so now I have it here and I need to get it alone so
I need to move all of these guys to the other side so I'll get rid of the t2 out of the denominator first although it doesn't really matter what order you do this in so I'll multiply both sides by t2 and now I have t2 on the bottom and t2 multiplied it on the top and now I have t2 times p1 times v1 equals p2 times v2 times t1 getting so close finally I can divide p2 times v2 to get them out of here to get t1 by itself do that to the other side P 2 / P 2 they cancel out V 2 over V 2
they cancel out and I'm going to do this in two steps I obviously have gotten rid of these guys so I could write it as this equals t1 but just to show you that I can flip it I'm going to write it as t1 equals t2 times p1 times v1 divided by p2 times v2 that's just if you prefer to have the variable alone that you're solving for on the left side just remember that you can always flip it so that's how we rearrange the

#### combined

gas well now I didn't do examples with all six
of these variables I only did it with four but based on the techniques that I use the shortcut technique at first or this more traditional method you should be able to figure out how to solve for any of these any of these variables that you need to