High-demand algebraic models

Objective. Form and solve linked equations from unfamiliar contexts.

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Learn: the key idea

High-demand algebraic models require turning a wordy, unfamiliar context into two or more equations that can be solved together to answer a practical question. Identify the unknown quantities, assign letters, write one equation per piece of given information, then solve the resulting system using substitution or elimination. For example, if a rectangle's perimeter is 36 cm and its length is 4 cm more than its width, letting width be w gives 2(w + w+4) = 36, leading to w = 7.

Worked example

A ticket costs £x for an adult and £y for a child. Three adults and two children pay £37, while two adults and four children pay £38. Use this algebraic model to find x and y.

1 $Form equations: 3x + 2y = 37 and 2x + 4y = 38$
2 $Multiply the first equation by 2: 6x + 4y = 74$
3 $Subtract the second equation: 6x + 4y - (2x + 4y) = 74 - 38, so 4x = 36, x = 9$
4 $Substitute into 2x + 4y = 38: 18 + 4y = 38, so y = 5$

Answer: x = £9, y = £5

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Common mistake. A common slip is writing an equation that only uses some of the given information, leaving the system underdetermined with too many unknowns. Always check that the number of independent equations matches the number of unknown letters before solving.

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