Consider the equation (x – 9)^{2} = x – 7. Is x = 11 a solution to this equation?

First evaluate the left-hand side expression for x = 11:

(11 – 9)^{2} = (2)^{2} = 4

Then evaluate the right-hand side expression for x = 11:

11 – 7 = 4

Because the left-hand and right-hand expressions match for x = 11, 11 is a solution to the equation. We can hang onto this one; we don't need to recycle it. Which is a shame, because we could've gotten 10 cents for it down at the grocery store.

Example 2

Is y = 3 a solution to the equation y + 2 = 2y?

First evaluate the left-hand side expression for y = 3:

3 + 2 = 5

Then evaluate the right-hand side expression for y = 3:

2(3) = 6

Since 5 definitely isn't equal to 6, the left- and right-hand sides of this equation don't give us to the same number when y = 3. So nope, y = 3 is not a solution to the equation y + 2 = 2y. Adios, y = 3. Go peddle your falsehood to some other sucker.

Be Careful: Work with the left and right sides of the equation separately. Otherwise, you'll be in danger of writing false equations or making a wrong turn. You don't want to make a wrong turn, because then it'll take your GPS like a minute and a half to recalculate.

Example 3

Is x = 2 a solution to the equation x^{2} = 3x?

If we write down (2)^{2} = 3(2), we're lying. Sorry. We'll never do it again.

Let's figure out, without lying, if x = 2 is a solution to the equation.

First evaluate the left-hand side expression for x = 2:

(2)^{2} = 4

Then evaluate the right-hand side expression for x = 2:

3(2) = 6

Because 4 doesn't equal 6, x = 2 is not a solution to the equation x^{2} = 3x. Ah. It feels oh-so-much better to arrive at the answer honestly, doesn't it?