Systems Word Problems

If a problem has two unknowns, you need two separate facts about them.

The routine:
1. Define both variables in writing, with units.
2. Write one equation per fact given.
3. Solve by substitution or elimination.
4. Answer the actual question in a sentence.

Common patterns:

• Totals: a count equation plus a value equation. 12 coins worth $2.10 gives n + d = 12 and 0.05n + 0.10d = 2.10.
• Mixtures: a total amount plus a total concentration.
• Rates: distance = rate × time for each traveller.

Money problems are the classic trap. Counting coins and valuing coins are two different equations, and mixing them up produces nonsense.

Modelling with a system requires two independent relationships. Two restatements of the same fact produce a dependent system with no unique solution, which is a modelling error rather than an algebraic one.

The standard families each have a canonical structure. Count-and-value problems pair x + y = total count with ax + by = total value. Mixture problems pair volumes with c₁V₁ + c₂V₂ = c_f V_f for concentrations. Uniform-motion problems build each row from d = rt, with the relationship between the two rows (equal distances, or distances summing to a total) supplying the second equation.

Units are the strongest check available: every term in an equation must share units, so an equation adding coins to dollars is malformed by inspection.

Finally, the algebraic solution must be tested against the context. A negative count or a fractional person indicates a setup error even when the arithmetic is flawless.