After solving a contextual system, test every answer against the situation. An answer that satisfies the equations but breaks a real-world condition is rejected, with a reason written down.
This lesson is part of systems of equations. It builds on linear and nonlinear systems.
What should I check?
After the algebra, run through this checklist.
- Is each length, area, price or time positive?
- Must the value be a whole number, such as people or tickets?
- Does the answer satisfy every condition in the wording, including words like “greater than”?
- Are the units stated in the final sentence?
Worked example 1: a garden
The length of a rectangular garden is 3 m more than its width, and its area is 70 m². Find the dimensions.
Let the width be w m and the length l m. Then l − w = 3 and lw = 70. Substitute l = w + 3: w(w + 3) = 70, so w² + 3w − 70 = 0, which factorises as (w + 10)(w − 7) = 0.
So w = −10 or w = 7. A width cannot be negative, so w = −10 is rejected. The width is 7 m and the length is 7 + 3 = 10 m.
Check: 10 − 7 = 3 and 10 × 7 = 70.
Worked example 2: a number problem with a stated condition
Two positive integers differ by 4 and their product is 96. Find them.
Let the smaller be y and the larger x. Then x − y = 4 and xy = 96. So y(y + 4) = 96, which gives y² + 4y − 96 = 0, or (y + 12)(y − 8) = 0.
The roots are y = −12 or y = 8. The numbers must be positive, so y = −12 is rejected, and y = 8 with x = 12. Check: 12 − 8 = 4 and 12 × 8 = 96.
The rejected root would give x = −8, so the pair (−8, −12) also fits the equations. The word “positive” in the question rules it out.
The mistake that costs marks
The common slip is to write both answers as final. The working is correct, but the last line ignores the situation.
| Step | Wrong | Right |
|---|---|---|
| Roots of the quadratic | w = −10 or w = 7 | w = −10 or w = 7 |
| Final answer | Width is −10 m or 7 m | w = −10 rejected, width cannot be negative |
| Dimensions | Two sets given | Width 7 m, length 10 m |
Always end with a sentence in words that names the quantity and its units.
Check yourself
The sides of a square are each increased by 4 cm in one direction and decreased by 1 cm in the other, forming a rectangle of area 50 cm². Find the side of the square.
Answer
Let the side be x cm. The rectangle has sides x + 4 and x − 1, so (x + 4)(x − 1) = 50.
Expand: x² + 3x − 4 = 50, so x² + 3x − 54 = 0, which factorises as (x + 9)(x − 6) = 0.
So x = −9 or x = 6. A length cannot be negative, so x = −9 is rejected. The side is 6 cm. Check: the rectangle is 10 cm by 5 cm, with area 50 cm².
What to study next
Test the whole chapter with the systems of equations practice set. The word-problem structure worksheet helps you organise a long question before you write any equation.
If you want a teacher to work through word problems with you, see online one-to-one Additional Mathematics tuition.