Solve the Rational Equation

I think when students see a problem like this they just see so many fractions and they immediately just say you know what this is going to be too much trouble I don't even want to try this I'll just skip to the next question let's not do that let's go ahead and think about what we need to be able to do to be able to solve this relatively easily now to preface that all I simply want to do is just create a very basic equation so you can see where my thought process is going to be going all right so there's two things I want you to understand if I had a problem like this like 12x equals a 5 Hales this had a has a lot of fractions too but notice here I have two in the denominator I want to get rid of that so how can I easily get rid of that well I can multiply everything by two when I multiply everything by two notice these denominators evenly divide into that two so therefore I'm just left with an x + 5 right I'm sorry x equal 5 let's do that there you go so you can see that problem actually wasn't really that bad multiplying by two gets rid of that denominator but what about if I had something like this X over 5 equals a 5es now again you could think about the cross product I don't want to go there right now I just want you to think now what I need to do is I can't just pick two because even though two divides into two like what do I need to multiply this by I don't want to just multiply by two right because five does not evenly divide by two however if I multiply it by two and a five or two times five then my five will divide into my five and my two will divide in my two that is what we call the least common multiple or the least common denominator when you're dealing with your denominators here so you always want to be able to identify that least common denominator when you have multiple fractions now in this case you can see that's going to be 10 and when I multiply 10 here on both sides I'm going to get a 2X is equal to a 25 and now you can go ahead and solve but notice how this gave me an equation without fractions over here for this problem that's exactly what I want to do now notice I do have an X and that is going to be like the third addition here that we have so hopefully you recognize if I want to get the rid of the two and the five cuz all my fractions either have a two or five I need to multiply by 10 furthermore I also have an x on the Teter I have to get rid of that X so I also have to multiply everything by an X okay so this is going to be kind of like my fraction Buster I'm going to multiply this by every single term if you don't multiply by every single term then you're not producing what we call an equivalent equation and you have to be able to do that right remember like the very basics of equations whatever you do on one side you have to do on the other side well when you're multiplying here by the scalar and you have terms separated by addition or subtraction or on both sides equations you have to make sure you multiply everything by there now rather than multiplying the numerators and then dividing the denominator in there I am going to do it a little bit quicker I going to divide first and then multiply so here I can say a 5x actually you know what let's just distribute this just so you can kind of see I will break it down we'll do it a little bit the slower way so you guys can see what's happening so I have a 10x is being multiplied by a 2 over a 5x minus Distributing that 10x * a 1 over a 2X right because we already have the minus and that's going to be equal to a 10x * a -2 okay so what I was trying to show you here is 5X evenly divides into 10x 2 times 2 * 2 is going to give me a four over here 2 divides into -10 that's going to be neg five times the X's are going to divide out and what was I at with neg five minus a 5 and then over here I have two divides into a 10x that's going to be equal to a 5x okay so now 4 - 5 is going to be a - 1 = a 5x IDE 5 IDE 5 x = A5 which is going to be did I forget to write the negatives I have a oh that's a negative oh I forgot to write this negative that should be a Nega five there we go so that's a negative negative so that's going to be a positive which is answer B