Is the equation 𝑥 squared plus 𝑦
squared over 𝑥 plus 𝑦 equals 𝑥 plus 𝑦 an identity?
Remember, an identity is an
equation that’s true for all values of our variable. So for this equation to be an
identity, 𝑥 squared plus 𝑦 squared over 𝑥 plus 𝑦 must be equal to 𝑥 plus 𝑦 for
all values of 𝑥 and 𝑦. One method we have is to manipulate
this fraction and see if we can simplify it. The problem is, we would usually
factor the expressions on the numerator and/or the denominator to do so. And these aren’t factorable. Now, this might be a hint that the
expression on the left is not equal to that on the right for all values of 𝑥 and
𝑦. And since an identity is true for
all values of 𝑥 and 𝑦, if we can find just one set of values where this equation
doesn’t hold, then we can show it’s not an identity.
Let’s try letting 𝑥 be equal to
one and 𝑦 be equal to two. Then, the expression on the left
becomes one squared plus two squared over one plus two, which is equal to
five-thirds. The expression on the right,
however, is simply one plus two, which is equal to three. It’s quite clear to us that
five-thirds is not equal to three. And so we found a value of 𝑥 and
𝑦 such that this equation doesn’t hold. It, therefore, cannot be an
identity; it doesn’t hold for all values of 𝑥 and 𝑦.