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# Creating an equation with infinitely many solutions

Video transcript

We're asked to use the drop-down
to form a linear equation with infinitely many solutions. So an equation with
infinitely many solutions essentially has the same
thing on both sides, no matter what x you pick. So first, my brain just wants
to simplify this left-hand side a little bit and
then think about how I can engineer the
right-hand side so it's going to be
the same as the left no matter what x I pick. So right over here,
if I distribute the 4 over x minus
2, I get 4x minus 8. And then I'm adding x to that. And that's, of course, going
to be equal to 5x plus blank. And I get to pick
what my blank is. And so 4x plus x is 5x. And of course, we
still have our minus 8. And that's going to be
equal to 5x plus blank. So what could we make
that blank so this is true for any x we pick? Well, over here we have
5 times an x minus 8. Well, if we make this a minus
8, or if we subtract 8 here, or if we make this
a negative 8, this is going to be true for any x. So if we make this
a negative 8, this is going to be true
for any x you pick. You give me any x, you multiply
it by 5 and subtract 8, that's, of course, going to
be that same x multiplied by 5 and subtracting 8. And if you were to try to
somehow solve this equation, subtract 5x from both sides,
you would get negative 8 is equal to negative 8, which is
absolutely true for absolutely any x that you pick. So let's go-- let me actually
fill this in on the exercise. So I want to make 5-- it's
going to be 5x plus negative 8.