Year 10: Linear equations with variables on both sides

When an x (or any variable) appears on both sides of an equals sign, it can look intimidating. The trick is to tidy the equation step by step until the variable lives on one side only. Follow the method below and you’ll turn any “messy” equation into a straightforward one‑step problem.

1. What does “variable on both sides” mean?

Equations like $5 x + 7 = 3 x - 9$ or $4 (2 - y) = 6 y + 10$ contain the unknown on each side of the equals sign. Your mission is to combine all the variable terms on one side and all the constants on the other.

Exercise

2. Expand any brackets first

Clear the decks by removing brackets using the distributive law. For example, in $4 (2 - y) = 6 y + 10$ :

 $4 \times 2 - 4 \times y = 6 y + 108 - 4 y = 6 y + 10$

Exercise

3. Collect like terms

Before you start shifting terms across the equals sign, combine any that already live on the same side. For $2 x + 4 - x = 3 x - 1$ , tidy the left side:

 $(2 x - x) + 4 = 3 x - 1 x + 4 = 3 x - 1$

4. Move the variable terms to one side

Pick the side with the smaller coefficient (often avoids negatives). Using $x + 4 = 3 x - 1$ :

 $Subtract x from both sides : x + 4 - x = 3 x - 1 - x 4 = 2 x - 1$

5. Move the constants to the opposite side

Now isolate the term containing the variable:

 $Add 1 to both sides : 4 + 1 = 2 x - 1 + 15 = 2 x$

6. Isolate the variable

Finish by dividing (or multiplying) to leave the variable alone:

 $Divide both sides b y 2 : 5 \div 2 = 2 x \div 2 x = 2.5 (or 5 / 2)$

7. Check your solution

Substitute back into the original equation to verify:

 $Left : 2.5 + 4 = 6.5 Right : 3 (2.5) - 1 = 7.5 - 1 = 6.5 ✓$

8. Common mistakes

Forgetting sign changes: When you subtract or add a term to both sides, copy the sign correctly. A missing negative flips answers.
Moving only the variable letter: Shift the whole term (e.g., subtract $3 x$ , not just $x$ ).
Dropping brackets too late: Expand early so you don’t leave stray multipliers attached to the variable.
Skipping the check: Plugging the answer back in catches arithmetic slips before they cost marks.

Exercise

9. Why mastering this matters

Linear equations with variables on both sides pop up in physics formulas, financial modelling, and even coding loops. Nailing the technique now builds algebraic muscle you’ll rely on for quadratic equations, simultaneous systems, and beyond. Remember: tidy, shift, isolate, check—then celebrate a balanced equation!

Practice!

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