The unknown on both sides

Two shelves hold the same number of pencils. Every box holds the same count. Shelf A has 5 boxes and 3 loose pencils. Shelf B has 2 boxes and 18 loose pencils. Call the pencils in one box k. How many pencils are in a box?

Lift the same boxes from both pans

Each box holds k pencils. Shelf A is 5k + 3. Shelf B is 2k + 18. The shelves match, so 5k + 3 = 2k + 18.

The letter k is on both sides. Lift 2 boxes from each pan. Both pans lose the same amount, so the beam stays level. Now k is on the left only. The left side is 3k + 3, because 5k − 2k is 3k. The right side is 18: 3k + 3 = 18.

Take 3 off both sides: 3k = 15, because 18 − 3 = 15. Share both sides by 3: k = 5, because 15 ÷ 3 = 5.

Check in the first line. Left: 5 × 5 + 3 = 28. Right: 2 × 5 + 18 = 28.

Rule: Take the smaller number of k's off both sides. If the counts differ, k then sits on one side.