Word Problems with Simultaneous Equations

The hardest part of a simultaneous-equations word problem isn’t the solving, it’s the translation: turning a sentence about a rectangle, a number pair or a curve into two equations. Once you have them, the substitution routine takes over.

The translation routine

1. Name the unknowns with letters (say $l$ and $w$).
2. Turn each fact into an equation. “Perimeter is 20” and “area is 24” become two equations.
3. Solve the resulting linear-and-quadratic pair.
4. Answer the actual question, and reject impossible values (negative lengths).

A worked example

A rectangle has a perimeter of $20$ cm and an area of $24$ cm². Find its length and width. Let length $l$, width $w$. Perimeter: $2(l + w) = 20 \Rightarrow l + w = 10$. Area: $lw = 24$. From the first, $l = 10 - w$. Substitute: $(10 - w)w = 24 \Rightarrow 10w - w^2 = 24 \Rightarrow w^2 - 10w + 24 = 0$. Factorise: $(w - 4)(w - 6) = 0 \Rightarrow w = 4$ or $w = 6$. The pair is $4$ cm and $6$ cm (the two roots are the width and length).

Notice the two roots give the two dimensions, a common, elegant feature of these problems.

Reject the impossible

Word problems live in reality, so negative or nonsensical solutions must be rejected, in writing. A length can’t be negative; a number of people can’t be fractional. Stating the rejection with a brief reason is often a mark.

Common mistakes

• Mis-translating the perimeter (using $l + w = 20$ instead of $l + w = 10$).
• Forgetting to answer the question asked (giving $w$ when both dimensions are wanted).
• Keeping an impossible negative solution.

Full topic context: Simultaneous Equations notes, and the worked product example.