a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Students should know that ultimately, we can reduce one-variable equations to three forms, each of which means something different about the possible answers.

If we reduce an equation into the form x = a (where a is some constant), then we only have one answer. For this problem, the right answer is a and there's no alternative. Math is as it should be. For instance, if we have x + 8 = 10, we can find that x = 2. We can't have any other number as x because 2 is the only number that will work for the equation we're given.

However, not all equations are that black-and-white. The equation 2x = 3x – x can be reduced to 2x = 2x. If we divide both sides by 2x, we get 1 = 1. What are we supposed to do with that? Well, we've reduced it into a form a = a (in this case, a = 1).

Basically, this means that the equation at hand is true when x is any real number. No matter what x is, 1 will always equal 1. So the equation will still be true when x = 3 and when x = 327,301.

The last possibility is that we get some sort of nonsense statement like 10 = 0. It doesn't take a genius to know that 10 and 0 aren't the same thing, so what kind of equation would give us an answer like that? We're guessing something like 2x + 10 = 2x. If we subtract 2x from both sides, we'll get 10 = 0. Okay, so what does that mean?

Students should know that if our answer takes the form of a = b where a and b are different, the equation has no solutions. Essentially, this means there is no value of x that will make the equation true. Ever. No matter what real number we plug in for x, the equation 2x + 10 = 2x will never be true because 10 will never equal 0, no matter how hard we wish it.

Students should be able to reduce any linear one-variable equation into one of these three forms and interpret the result.