SL 1.6—Simple proof

An equality ($=$) states that two expressions have the same value for some values of a variable (e.g. $x^2=9$ is only true for $x=3$ or $x=-3$). An identity ($\equiv$) states that two expressions are equal for all values of the variable.

The correct description is: $=$ holds for specific values, $\equiv$ holds for all values. For example, $(a+b{)}^2\equiv a^2+2ab+b^2$ is true for every real $a$ and $b$, while $x+3=7$ is only true for $x=4$.

The option stating '$=$ means true for specific values, while $\equiv$ means true for all values' is precisely the correct distinction.

The question asks for the correct distinction between the equality symbol $=$ and the identity symbol $\equiv$.

The option '$=$ means true for all values, while $\equiv$ means true for specific values only' has the definitions completely backwards — this is a common misconception and is incorrect.

'Both $=$ and $\equiv$ mean the same thing' is incorrect — they carry distinct mathematical meanings. The $\equiv$ symbol specifically signals an identity that holds universally.

'$\equiv$ is only used in geometry' is false — $\equiv$ is used in algebra to denote identities, not restricted to geometry.

The remaining option — '$=$ means true for specific values, while $\equiv$ means true for all values' — is the correct answer.

A valid LHS to RHS proof requires starting with one side only — typically the more complex side — and transforming it step by step until the other side is reached.

The LHS, $(x+5{)}^2-25$, is more complex. We should start there and expand: $(x+5{)}^2-25=x^2+10x+25-25=x^2+10x$.

The option that begins with $(x+5{)}^2-25$ and correctly expands the bracket before subtracting 25 is the valid proof method.

The question asks which starting point leads to a valid LHS to RHS proof of the identity.

Writing $x^2+10x=(x+5{)}^2-25=x^2+10x$ assumes what is being proved — this is circular reasoning and not a valid proof.

Substituting $x=1$ into both sides only verifies the identity for one value. This is not a proof — it does not establish the result for all values of $x$.

Starting with $x^2+10x$ (the simpler RHS) and factorising is less natural and harder. The preferred method starts from the more complex LHS.

The correct approach begins with $(x+5{)}^2-25$, expands to get $x^2+10x+25-25$, and simplifies to $x^2+10x=$ RHS.