Equation with the variable in the denominator

so I have the equation 7 minus 10 over X is equal to 2 plus 15 over X and so this isn't the type of equation that you might think that you're used to solving but I'll give you a little a few moments to see if you can solve it on your own well what we'll see is we can do a quick multiplication of both sides to actually simplify this to a form that we are more used to looking at so what's probably bothering you because it's bothering me is these X's that we have in the denominators right over here we're like well how do we deal with that well whenever we see an X in the denominator the temptation is to multiply it by X but we can't just multiply one of the terms by X we have to multiply the entire side by X so we could multiply this entire side by X but we can't just multiply the left-hand side by X we'd also want to multiply the right-hand side by X and so what will that give us well we distribute the X we get x times 7 is 7x and then x times negative 10 over X well that's just going to be negative 10 so you get negative 10 right over there so the left-hand side simplifies to 7x minus 10 and then your right-hand side once again distribute the X x times 2 is 2x x times 15 over X well x times something over X is just going to be the something x times 15 over X is just going to be 15 plus 15 so now we've simplified this to a linear equation we have the variable on both sides so we just have to do some of the techniques that we already know so the first thing that I like to do is maybe get all my X's on the left-hand side so I want to get rid of this 2x right over here so I subtract 2x from the right-hand side now and I always remind you I can't do that just to the left just to the right hand side if I did just to the right hand side it wouldn't be an equality anymore you have to do that to the left hand side as well and so we are left with we are left with let me get that pink color again the left hand side 7 X 7 X 7 of something minus 2 of something well you're going to have 5 of that some - ten these two exes negate each other and you're left with equals 15 now we can get rid of this negative ten by adding 10 to both sides now like that green color when I do stuff to both sides so I can add 10 to both sides and I am left with 5x these negate each other is equal to 25 and this is the homestretch you see where this is going you can divide both sides by 5 and we are left with X is equal to 5 now let's verify that this actually worked so let's go back to the original equation we have 7 minus 10 over 5 this needs to be equal to I'm just taking our 5 and substituting it back here this needs to be equal to 2 plus 15 over 5 so this is 7 minus 10 over 5 this is just 2 needs to be equal to 2 plus 15 over 5 which is just 3 so 2 plus 3 7 minus 2 is 5 2 plus 3 is 5 5 is indeed equal to 5 and we are done