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G2.9 Solving One-Step Equations

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01 — Explanation

What does it actually mean to "solve" an equation?

An equation states that two expressions are equal; solving it means finding the value of the variable that makes this true. The equals sign is not an "answer button" — it's a statement of balance, like old-fashioned weighing scales. Whatever you do to one side, you must do identically to the other, or the balance is destroyed.

02 — Explanation

How do I use inverse operations to isolate the variable?

Identify what operation is being done to the variable, then apply its inverse to both sides: addition undoes subtraction (and vice versa), multiplication undoes division (and vice versa).

Equation	Operation on variable	Inverse applied	Solution
$k+18=21$	$+18$	$-18$ from both sides	$k=3$
$x-7=12$	$-7$	$+7$ to both sides	$x=19$
$2x=18$	$\times2$	$\div2$ both sides	$x=9$
$\tfrac{x}{4}=6$	$\div4$	$\times4$ both sides	$x=24$

Common Mistake

given $x-4=13$ , some students subtract 4 from both sides instead of recognising that the inverse of $-4$ is $+4$ . Read the sign as part of the term: it tells you which inverse to use, not which operation to repeat.

03 — Explanation

How do I check my solution is correct?

Substitute your answer back into the original equation and confirm both sides match. For $2x=18$ with $x=9$ : $2\times9=18$ ✓. This single habit catches almost every arithmetic slip before you move on.

Exam Tip

"Without sufficient working, correct answers may be awarded no marks" is explicit Edexcel guidance. A method mark (M1) is given for correctly applying the inverse operation even if an arithmetic error follows — so always show the step where you act on both sides, not just the final value.

04 — Worked example

Worked exam-style question

A one-step equation appears as the final stage of a longer geometry problem: having simplified an expression for a missing angle, a student arrives at $5y - 16 = 29$ ... but suppose instead the simplified equation is simply $5y = 45$ . Solve for $y$ and verify your answer.

The variable is multiplied by 5; the inverse is division.
Divide both sides by 5: $\dfrac{5y}{5} = \dfrac{45}{5}$ , giving $y=9$ .
Check: $5 \times 9 = 45$ ✓.

Answer: $y = 9$ .

Exam Tip

one-step equations rarely appear in isolation — they're usually the last line of a longer problem (angles, perimeters, word problems). Make solving them fast and automatic so you can focus your attention on setting the equation up correctly in the first place.

05 — Explanation

Why do some students get this wrong even when they "know the method"?

The usual cause isn't the algebra itself but the equals sign: in arithmetic it signals "here's the answer", but in algebra it signals "these two things balance". A student who hasn't made that mental shift will often perform an operation on only one side, destroying the balance. Explicitly thinking "whatever I do to the left, I must do to the right" before writing anything down prevents this.