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# Solving linear equations and linear inequalities — Harder example

## Video transcript

in the equation shown above a is a constant for what value of a does the equation have infinitely many solutions so you end up with infinitely many solutions if your equation simplifies to something like X is equal to X or one is equal to one something that's true that's going to be true for any X that you pick so let's let's see what we could do with this thing right over here and these are obviously not you know if you got a hundred equals 100 that would be the same that would have an infinitely many solutions zero equals zero these were all be situations where you have an infinite number of solutions so when I look at this thing up here my first instinct is well let's just see if I can simplify this a little bit I'll leave the a in there and then see if I can get to a point where it's going to have an infinite number of solutions so let me just rewrite it so we're gonna have 3 plus 10 X minus 5 is equal to a plus 1 times X minus 2 so let's see on the left-hand here I can add the 3 and the negative 5 or I could take 3 minus 5 that would be negative 2 so I get 10 X minus 2 is equal to and let me distribute the X so it's going to be a X plus X all I did here is I distributed I distributed the X minus 2 now let's see what happens if let's see I could get rid of both of these negative 2 is if I add 2 to both sides so if I just remember anything I do to one side I've got to do to the other one if I want to hold the Equality to continue to be true so I just added 2 to both sides and I am left with 10 X is equal to ax plus X let's see let's subtract X from both sides so if I subtract X actually I can write it like this I could subtract X from both sides on the left hand side I'm gonna get 9 X on the right hand side I'm going to get a a X so how could I have an infinite number of solutions an equation that's going to be true for any X well if a was equal to 9 because if a is equal to 9 I'm gonna have a situation so if a is equal to 9 then you're gonna have a situation where 9x is going to be equal to instead of a odd right 9 is going to be equal to 9x well that's going to be true for any X any x times 9 is going to be equal to that same x times 9 again you're gonna have an infinite number of solutions and so a needs to be equal a it needs to be equal to a is equal to 9 now what's really interesting here is think about what happened if a is any of these if a is any of these other things right over here then you're going to force a different solution but anyway we'll we'll leave that for for another another video