L4 · Variables on both sides

Learning goal

“I can solve equations with the variable on both sides by collecting the variable terms on one side and the constants on the other - including equations that also need conversion (L2) or expansion (L3) first - and verify the solution against BOTH sides of the original equation.”

Variable term — a term containing the variable, like 5x or (3/4)x, wherever it appears in the equation.

Constant term — a term with no variable, like +3 or -8.

Collect — to gather all variable terms onto one side of the equation and all constant terms onto the other, using preservation of equality.

Equivalent equation — same solution, new clothing (review from L2): every collecting move produces one.

Worked example

Solve 5x + 3 = 2x + 12. Step 1: Picture the balance: five bags and 3 loose units on the left pan; two bags and 12 loose units on the right. Step 2: Remove 2 bags from BOTH pans (preservation of equality): 3x + 3 = 12. Step 3: Remove 3 loose units from both pans: 3x = 9. Step 4: Divide both sides by 3: x = 3. Step 5: Verify against BOTH sides of the original: left 5(3) + 3 = 18; right 2(3) + 12 = 18. The two sides agree - that agreement is the whole point of the form. Say it plainly: variables on both sides is not a new kind of equation. It is the same balance with bags on both pans: unload the smaller bag-pile from both sides, then solve what remains.

Check yourself — explain it

Why do we subtract 2x - the smaller variable term - rather than 5x?

Paired problem — example still visible

Solve 6x + 1 = 3x + 10 -> x = 3 (subtract 3x: 3x + 1 = 10; 3x = 9; check: 6(3) + 1 = 19 and 3(3) + 10 = 19)

Faded completion — last step is yours

"7x - 2 = 4x + 7: subtract 4x from both sides to get 3x - 2 = 7, then 3x = 9, so x = ___" -> 3

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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