To show an equality is true, you need to use mathematical reasoning and proof techniques. This involves manipulating the given equations or expressions to show that they are equivalent, and then using logical steps to show that they are indeed equal.
Some common proof techniques used to show equality include substitution, algebraic manipulation, and using properties of equality such as the reflexive, symmetric, and transitive properties. Induction and contradiction are also commonly used for more complex proofs.
For example, to show that (x + 3)(x - 3) = x^2 - 9, we can expand the left side using the distributive property to get x^2 - 3x + 3x - 9. Then, we can combine like terms and see that it simplifies to x^2 - 9, proving that the two expressions are equal.
Mathematical notation such as equals signs (=) and logical symbols (<, >, ≤, ≥) can be used to show the steps in a proof for an equality. For example, if we are using substitution to show that a = b, we can write a = c and b = c, and then use the transitive property to show that a = b.
Yes, some tips for successfully showing an equality is true include carefully examining the given expressions or equations, breaking them down into simpler components, and using known properties and techniques to manipulate them into an equivalent form. It is also important to clearly explain each step and justify why it is valid in the context of the proof.