What is the difference between an equation and an identity?
Step-by-step solution
To find: How an identity differs from an ordinary equation
Idea: Both have an equals sign. The test is: does it stay true whatever numbers you put in? If yes for every value, it is an identity. If only for some values, it is just an equation, and those values are its solutions.
- Definition. An algebraic identity is an equation that is true for all values of the variables in it. An equation need not be true for all values; it may be true only for particular values.1 mark
- Example of an equation: x2 − 1 = 24. This means x2 = 25, so it is true only for x = 5 or x = −5. Put x = 1: 1 − 1 = 0 ≠ 24. So it is an equation, not an identity.½ mark
- Example of an identity: (x + y)2 = x2 + 2xy + y2 is true for all values of x and y (try x = 3, y = −1: 4 = 9 − 6 + 1). So this equation is also an identity.½ mark
Check: To show something is NOT an identity, one value that fails is enough (x = 1 fails x² − 1 = 24). To show it IS an identity, you must prove it in general (for example by expanding), not just test a few values.
Answer to write in the exam
Identity: an equality true for all values of the variables, e.g. (x + y)2 = x2 + 2xy + y2
Equation: an equality true only for some values of the variables, e.g. x2 − 1 = 24 is true only for x = 5 or x = −5
∴ Every identity is an equation, but an equation is an identity only if it holds for all values of its variables.
Common mistakes that cost marks
- Thinking an identity is something different from an equation. An identity is a special kind of equation, one that is always true.
- Calling something an identity after checking only one or two values. A few checks can agree by luck; a general proof is needed.
- Saying x2 − 1 = 24 has only the solution x = 5. (−5)2 = 25 too, so x = −5 also works.
How this can come in the exam
Which of the following is an identity?
- 2x + 3 = 11
- x2 = 9
- (a − b)2 = a2 − 2ab + b2
- x2 + 1 = 2x
Show answer
(C) (a − b)2 = a2 − 2ab + b2
(a − b)2 = a2 − 2ab + b2 holds for all a, b. The others hold only for x = 4, x = ±3 and x = 1 respectively.
Assertion (A): (a + b)2 = a2 + b2 is not an identity.
Reason (R): For a = 1, b = 1, the left side is 4 and the right side is 2.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
One pair of values that fails is enough to show a statement is not an identity, so R explains A.
Try one yourself
State whether each is an identity or only an equation: (a) x(x + 2) = x2 + 2x (b) x(x + 2) = 8.
Show answer
(a) Identity: the distributive property gives x2 + 2x for every x. (b) Equation: true only for x = 2 or x = −4 (2 × 4 = 8, (−4) × (−2) = 8); for x = 0 it gives 0 ≠ 8.
More questions like this
- By now you must have observed that (a + b)2 ≠ a2 + b2. But can you find out which one of them will be greater? For example, if a = 10 and b = 2, what are the values of (a + b)2 and a2 + b2?
- 1. What can you say about a and b if (a + b)2 < a2 + b2?
2. What can you say about a and b if (a + b)2 > a2 + b2?
3. When will (a + b)2 be equal to a2 + b2?
Did you observe that (a + b)2 and a2 + b2 are both positive? What term will decide which is larger? Use the expansion of (a + b)2 to decide. - Let us try to expand (5x + 2y)2.
- To calculate 432, we can write it as (40 + 3)2.
- Using the identity (a + b)2 = a2 + 2ab + b2, expand the following: