• Free
  • Grade 4–6
  • Challenge · totals to 80c, 120c or 160c

One equation, two unknowns worksheets

Pens are 15c, erasers are 8c, and Mia spent exactly 115c on both. As algebra that has infinitely many answers. Here it has exactly one, and the reason is that nobody buys two thirds of a pen.

  • Answer key included
  • Reopen this exact sheet
  • Letter & A4 ready
Ready to printOne equation, two unknowns
One equation, two unknowns worksheet — Challenge · totals to 80c, 120c or 160c, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

This is the sheet where being a whole thing does real mathematical work. One equation with two unknowns is normally underdetermined — a line of solutions, no single point. Add the requirement that both counts are whole numbers and the line has just one point on it with integer coordinates. A child does not need the vocabulary to use the idea, and using it before naming it is the right order.

Why this sheet works

The method is trial with a reason attached: whatever is left after the dearer item has to divide exactly by the cheaper price. So you try one of the dearer item, then two, then three, and reject each one because the remainder will not divide. The answer key prints those rejections rather than only the answer, because the rejected trials are where the thinking is — a child who arrived by luck and one who eliminated four possibilities deserve different feedback.