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