Year 10: Solving linear equations

Solving equations can feel like detective work: you’re given a mathematical mystery and your job is to uncover the value of the unknown that makes everything true. In Year 10, mastering this skill is essential because it underpins topics from algebra and functions to physics and finance problems you’ll meet in senior school.

1. Keep the balance

An equation is like a perfectly balanced scale. Whatever you do to one side, you must do to the other to keep the balance. That single idea powers every method that follows, so keep it front‑of‑mind.

Here is an example of an equation as a scale

2. One‑step and two‑step equations

Start simple. For $x + 7 = 23$ , subtract 7 from both sides to get $x = 16$ . For $5 x = 35$ , divide both sides by 5. Two‑step equations combine these moves. In $3 x - 4 = 17$ , first undo the subtraction (add 4), then undo the multiplication (divide by 3) to reveal $x = 7$ .

Exercise

3. Distributive law and collecting like terms

Equations rarely arrive tidy. Suppose you have $4 (2 - y) = 3 y + 10$ . Expand the bracket: $8 - 4 y = 3 y + 10$ . Next, gather the y terms on one side by adding 4y: $8 = 7 y + 10$ . Then isolate the variable just as before: subtract 10, giving $- 2 = 7 y$ , and finally divide by 7: $y = - 2 / 7$ .

Review on multipliation laws

Exercise

4. Variables on both sides

When letters appear on each side, move the smaller collection to one side to avoid negatives later. For $6 p + 9 = 4 p - 5$ , subtract 4p from both sides: $2 p + 9 = - 5$ . Now subtract 9: $2 p = - 14$ , then divide by 2: $p = - 7$ . Always keep the scale balanced—move whole terms, not just the letters.

Exercise

5. Check your answer

Substituting back is the algebraic equivalent of proofreading. Plug $p = - 7$ into the original: left side 6(–7)+9 = –42+9 = –33; right side 4(–7)–5 = –28–5 = –33. Both match, so the solution is sound. Checking catches sign slips and arithmetic errors before they snowball.

6. Common trip‑ups (and how to dodge them)

Forgetting to distribute a negative sign: remember that $- (3 x - 2)$ becomes $- 3 x + 2$ .
Dividing too early: clear brackets and collect like terms first; otherwise you might split terms that belong together.
Mixing unlike terms: $5 x$ and $3 x$ combine, but $5 x$ and $3$ do not.

7. Why it matters

Solving equations isn’t just a syllabus dot point. It trains logical thinking, precision, and perseverance—qualities that engineers, coders, and even game designers rely on daily. Next time you wrestle with an algebra question, remember you’re practising a universal problem‑solving language. Grab some past‑paper questions, apply the balance rule step by step, and don’t skip the check. And if you get stuck, rewrite the question, annotate each step, and talk it through—explaining aloud often reveals where the hitch is hiding.

Here are some exercise quizzes to further sharpen your equation solving skills:

Challenge problems: