How Do You Write a Proof Going From the Left-Hand Side to the Right-Hand Side (LHS to RHS)?

Transform the left-hand side (LHS) one valid algebraic step at a time until it becomes the right-hand side (RHS). Begin with only the LHS, justify each transformation, and finish by stating that it equals the RHS.

The Method

An LHS-to-RHS proof establishes an identity, meaning the two expressions are equal for every value in their common domain. This is part of Mathematics: Analysis and Approaches, SL 1.6: Simple deductive proof, and is suitable for Paper 1 because it uses exact algebra rather than technology.

Use this structure:

1. Write “LHS” and copy the left-hand expression.
2. Apply a known algebraic rule, such as factorization, expansion, a common denominator, or an established identity.
3. Put each new equivalent expression on a separate line, joined by an equals sign.
4. Continue until the expression is exactly the RHS.
5. Finish with “$=\text{RHS}$” or “Therefore, LHS $=$ RHS.”

Why This Works

Each line must be equivalent to the previous line, so the chain preserves truth from the original LHS to the final RHS. For example, multiplying the numerator and denominator of a fraction by the same non-zero expression preserves its value. Factoring is also reversible, but cancelling a factor is valid only when that factor is non-zero.

Worked Example

Show that

$$\frac{1}{m+1}+\frac{1}{m^2+m}\equiv\frac{1}{m}, \qquad m\neq 0,-1.$$

Starting from the LHS,

$$\begin{aligned} \frac{1}{m+1}+\frac{1}{m^2+m} &=\frac{1}{m+1}+\frac{1}{m(m+1)} \\ &=\frac{m}{m(m+1)}+\frac{1}{m(m+1)} \\ &=\frac{m+1}{m(m+1)} \\ &=\frac{1}{m} \\ &=\text{RHS}. \end{aligned}$$

The domain restrictions are necessary because the original denominators cannot equal zero.

A common misconception is that you should manipulate both sides simultaneously. Do not do this: it assumes the equality you are trying to establish and may produce circular reasoning. Also avoid beginning with “LHS $=$ RHS,” because that is the conclusion.

Exam Technique

For the command term show that, examiners award marks for the logical working, not for repeating the given result. Display every important algebraic step, use $\equiv$ for an identity where appropriate, and never cancel a factor without checking when it could be zero.