If a is any of these, if a is any of these other Interesting here is think about what would happen And so a needs to be equal, a it needs to be equal Any x times nine is going to be equal to that same x times nine again. So if a is equal to nine then you're gonna have a situation where nine x is going to be equal to, instead of a, I'd write nine. That's gonna be true for any x? Well if a was equal to nine, because if a is equal to nine So how could I have an infinite number of solutions, an equation So if I just, remember anything I do to one side I've gotta do to the other one if I wanna hold the equality to continue to be true. Now let's see, what happens if, let's see, I could get rid of both of these negative twos if All I did here is Iĭistributed the x, minus two. So I get 10x minus two is equal to, let me distribute the x. So let's see, on the left-hand here I can add the three and the negative five. Plus 10x minus five is equal to a plus one times x minus two. I'll leave the a in there and then see if I can get to a point where it's gonna have an 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. These were all be situations where you have an infinite number of solutions. That would be the same, that would have an These are obviously not, if you got 100 equals 100, So let's see what we could do with this thing right over here.

That's going to be true for any x that you pick. Or one is equal to one, something that's true Infinitely many solutions if your equation simplifies to something like x is equal to x,

For what value of a does the equation have infinitely many solutions? So you end up with

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