how do you do equality proofs?
To do an equality proof, you typically need to show that two expressions or equations are equal using logical steps and properties. Here's a step-by-step guide on how to approach equality proofs:
1. Start with the given equation or expression that you want to prove equal.
Example: Prove that (a + b)^2 = a^2 + 2ab + b^2.
2. Identify the desired result. Determine what you need to prove the equation is equal to.
Example: The desired result is a^2 + 2ab + b^2.
3. Break down both sides of the equation or expression. simplify and manipulate each side separately using known mathematical properties.
Example:
(a + b)^2 = (a + b)(a + b) (using the formula for squaring a binomial)
= a(a + b) + b(a + b) (using the distributive property)
= a^2 + ab + ba + b^2 (expanding each term)
= a^2 + ab + ab + b^2 (simplifying)
= a^2 + 2ab + b^2 (combining like terms)
4. Make sure to include logical reasoning and known properties of mathematics. Each step needs to be justified.
Example: Each step is justified by applying known mathematical properties like the distributive property, commutative property, and associative property.
5. Compare the simplified version from step 3 to the desired result. If they are the same, you have successfully completed the equality proof.
Example: The simplified version is a^2 + 2ab + b^2, which matches the desired result.
Remember that in equality proofs, it's crucial to show the logical steps and provide a clear explanation for each manipulation. This ensures that your proof is valid and understandable to others.