Equations with Variables on Both Sides

🧱 See It

The balance scale still works exactly the same way. But sometimes mystery bags (x) show up on both pans at once. The fix: remove matching bags from both pans first — subtract the smaller variable term from both sides — until only one pan has bags left. After that, it solves exactly like a normal two-step equation.

✏️ Draw It

Solve 5x + 3 = 2x + 12. Both pans start with bags on them:

Move: remove 2 bags from both sides — the smaller bag count, so no side ever goes negative.

Only one pan has bags left: 3x + 3 = 12. From here it's an ordinary two-step equation.

🔢 Write It

5x + 3 = 2x + 12

Step 1 — subtract 2x from both sides:

3x + 3 = 12

Step 2 — undo the addition (subtract 3 from both sides):

3x = 9

Step 3 — undo the multiplication (divide both sides by 3):

x = 3

Subtract the SMALLER variable term from both sides first — this leaves only one variable term, on one side, and everything after that is a normal two-step equation.

💡 Worked Examples

Example 1: 4x − 5 = 2x + 7

Subtract 2x from both sides: 2x − 5 = 7
Add 5 to both sides: 2x = 12
Divide both sides by 2: x = 6
Verify: 4(6) − 5 = 19 and 2(6) + 7 = 19 ✓

Example 2: 3x + 8 = 7x − 4

The smaller variable term is 3x — subtract it from both sides: 8 = 4x − 4
Add 4 to both sides: 12 = 4x
Divide both sides by 4: x = 3
Verify: 3(3) + 8 = 17 and 7(3) − 4 = 17 ✓

🧠 Quick Reference

Subtract the SMALLER variable term from both sides first — this leaves only one variable term, on one side. Then solve like a normal two-step equation. Always verify by substituting your answer back into BOTH sides of the original equation.