How do you distinguish between an identity and an equation?

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SUMMARY

The discussion centers on distinguishing between identities and equations in mathematics, specifically addressing the expression x² + y² = 1. Participants clarify that identities, such as sin²x + cos²x = 1, hold true for all values of x, while equations like 3x² + x = 2 are conditional and true only for specific values. The use of symbols such as "≡" indicates an identity, whereas "=" denotes an equation. The conversation emphasizes the importance of context in determining whether a statement is an identity or an equation.

PREREQUISITES
  • Understanding of mathematical functions and their properties
  • Familiarity with the concepts of identities and equations
  • Knowledge of mathematical notation, including "=" and "≡"
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research the properties of mathematical identities and their applications
  • Explore conditional equations and their significance in problem-solving
  • Study the implications of using different mathematical symbols in expressions
  • Learn about quantifiers in mathematical statements, such as "for all" and "there exists"
USEFUL FOR

Mathematics students, educators, and anyone interested in deepening their understanding of mathematical statements, particularly in distinguishing between identities and equations.

  • #31


tahayassen said:
If ≡ was replaced with =, wouldn't it also be true that the coefficients of their respective terms are equal?

No, because the two equations would have different meanings. Let's look at the equation in my example a couple of posts ago.

As an identity:
2 ##\equiv## (A + B)x + A

As an identity, the equation above has to be true for all values of x. In my previous work I solved for the constants A and B, and found them to be A = 2, B = -2.

As an ordinary equation:
2 = (A + B)x + A
If we interpret the above as an ordinary equation (not an identity), given values of A and B, we could solve for the value of x that makes the equation a true statement.

The work would look like this.
2 - A = (A + B)x
$$\Rightarrow x = \frac{2 - A}{A + B}$$
If someone tells us the values of the constants A and B, we can determine the value of x.
 

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