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## Heart of algebra

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# Solving linear equations and linear inequalities — Harder example

## Video transcript

- [Voiceover] 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 see what we can do with this thing right over here. These are obviously, now, you know, if you've got 100 equals 100, now, that would be the same, that would have infinitely many solutions. Zero equals zero, these
would 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 re-write it. So, we're going to have three plus ten x minus five is equal to a plus one times x minus two. So, let's see, now, on the left-hand here, I can add the three and the negative five, or I can take three minus five,
that would be negative two. So I get ten x minus two is equal to, now, let me distribute the x. So, it's going to be ax plus x. All I did here is, I
distributed the x minus two. Now, let's see, what happens if, so, let's see, I could get rid of both of these negative twos, if
I add two to both sides. 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. So, I just added two to both sides, and I am left with ten x is equal to ax plus x. Now, let's see, let's
subtract x from both sides. So, if I subtract x, actually, I can just write it like this, I could subtract x from both sides. On the left-hand side,
I'm gonna get nine x. On the right-hand side,
I'm going to a, ax. So, how could I have an
infinite number of solutions an equation that's
gonna be true for any x? Well, if a was equal to nine,
because if a is equal to nine, I'm gonna have a situation. So, if a is equal to nine, then you're going to have
a situation where nine x is going to be equal to,
instead of a, I'd write nine, is going to be equal to nine x. Well, that could be true for any x. Any x times nine is going to be equal to that same x times nine again. You're going to have an
infinite number of solutions. And, so, a needs to be equal, a, it needs to be equal
to, a is equal to nine. Now, what's really interesting here is, think about what would
happen if is any of these other things right over here. Then, you're going to
force a different solution. But anyway, we'll leave
that for another video.