True for some, or for all?

The idea

An algebraic identity is an equation that is true for all values of the variables in it, while an equation need not be true for all values.

A letter in a statement

x² − 1 = 24 has a letter in it. For x = 5 it is true: 5² − 1 = 24. But what about other numbers?

Is x² − 1 = 24 true for every number x?

• Yes, for every x
• Only for some x
• For no x

Test them both

Card 1: slide x and find every number that makes it true. Card 2: choose x and y and compare the two sides.

Equation or identity

An identity is an equation that is true for all values of the variables in it. An equation need not be true for all values.

x² − 1 = 24 is true only for x = 5 or x = −5, so it is just an equation. (x + y)² = x² + 2xy + y² is true for all x and y, so it is also an identity.

Notes

An identity is an equation that is true for all values of its variables. An equation may be true only for some values.

Check yourself

Which of these is an identity?

x² − 1 = 24. Which positive number x makes it true?

Answer: 5

5² = 25 and 25 − 1 = 24, so x = 5. The other solution is x = −5.

For x = 1 and y = 1 the identity (x + y)² = x² + 2xy + y² gives 4 = 4. Does one example prove it is an identity?

Is 3(x + 2) = 3x + 6 an identity?

• x + 3 = 10. This is true only when x = 7.
• 2x = 8. This is true only when x = 4.
• x + x = 2x — correct. Yes. Whatever x is, x + x is 2x.
• No. An identity must work for every x and y — correct. Yes. One example can show that a statement is false, but only a reason that covers all numbers proves an identity.
• Yes, because the two sides agree. The two sides also agree for x = 5 in x² − 1 = 24, and that is not an identity.
• Only if we also try x = 2 and y = 3. Even many examples do not prove it. The square picture and the distributive property do.
• No, it is true only for x = 2. Try x = 1: 3 × 3 = 9 and 3 × 1 + 6 = 9. It is true there too.
• Yes, it is true for every x — correct. Yes. The distributive property gives 3(x + 2) = 3x + 6 whatever x is.
• No, it is true only for x = 0. Try x = 1: 3 × 3 = 9 and 3 × 1 + 6 = 9. It is true there too.