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Three-variable systems from a story worksheets

Three shopping trips, three totals, and three prices to find. These free printable worksheets ask a student to write a system of three equations from a story and then solve it, with an answer sheet that shows the working.

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Three-variable systems from a story
Three-variable systems from a story worksheet — Three prices, whole dollars, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Each line of the story becomes one equation. If 1 apple, 2 pens and 1 bagel cost $8, then a + 2p + b = 8, with one letter for each price. Writing those equations is half of the question, and it is the half that a page of ready-made systems never practices.

Why this sheet works

Many of these stories have a pleasant surprise in them. When two items appear in the same ratio on two lines, subtracting removes both at once and one price falls out immediately. Looking for that before starting saves a lot of work, and it is the kind of noticing that separates a careful solver from a mechanical one.

How this one works

1 muffin, 1 bagel and 2 peaches cost $12; 2 muffins, 1 bagel and 2 peaches cost $16; 3 muffins, 4 bagels and 2 peaches cost $26

(1) m + b + 2p = 12
(2) 2m + b + 2p = 16
(3) 3m + 4b + 2p = 26
(4) 2 × (1) − (2): b + 2p = 8
(5) 3 × (1) − (3): −b + 4p = 10
(6) [(4) + (5)] ÷ 6: p = 3
b = 2
m = 4
  1. Each line of the story is an equation, with one letter for each price.
  2. Take m out of two different pairs. That leaves two equations with only b and p in them.
  3. Now take b out of those two. Only p is left, so its value is one division away.
  4. Put p = 3 into (4) to find b, then both into (1) to find m.

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