When no number fits

Shelf A holds 2 boxes and 5 loose pencils. Shelf B holds 2 boxes like them and 9 loose pencils. Can both shelves hold the same number? Call the pencils in one box k.

Take the k's away

No. Shelf B has 4 more pencils than A, whatever a box holds.

Lift 2 boxes from each pan. Shelf A has 5 loose pencils left. Shelf B has 9. Now the line says 5 = 9.

That is false, whatever k is. An equation that no number makes true has no solution.

Compare 3k + 5 = 2k + 9. Take 2k off both sides: k + 5 = 9. Then k = 4, because 9 − 5 = 4. One number fits.

Compare 3k + 4 = 3k + 4. Take 3k off both sides: 4 = 4. That is true, whatever k is. Every number fits.

Different counts, such as 3k + 4 = 5k + 4, leave one number. Here k = 0. Take 3k and 4 off both sides to see why: 0 = 2k.

Plain numbers are numbers with no letter.

Rule: If both sides hold equal numbers of k's, take them off. Plain numbers that differ: no number fits. Plain numbers that match: every number fits.