Systems of linear equations — Mathematics, 14–17 years
How two conditions can meet at one pair of values, and how graphs and algebra find that meeting point.
Two conditions, one answer
A system is a set of equations that must be true at the same time. Its solution is the value, or pair of values, that satisfies every equation. On a graph, two straight lines usually meet at one point; that point gives the shared x and y values.
Why combine equations?
Many real questions contain two unknown amounts but give two clues about them: two ticket types, two prices, or two mixtures. One equation alone leaves many possibilities. Combining both clues removes the wrong possibilities and finds the values that fit the whole situation.
Tickets at a concert
A concert sells 10 tickets for €86. Adult tickets cost €10 and student tickets €6. Let a be adult tickets and s student tickets: a + s = 10 and 10a + 6s = 86. Replace s with 10 − a: 10a + 6(10 − a) = 86, so 4a = 26 and a = 6.5. Since tickets must be whole, these figures reveal that the stated total is impossible.
Checking matters
A tempting mistake is to stop after finding one variable and assume the job is finished. That is reasonable because the algebra may look complete, but a system requires both values and both equations. Substitute the answer back into each original equation; an unnoticed sign error will then be exposed.
Finding a hidden split
Systems help separate combined information: a shop can infer how many items of two prices were sold, or a scientist can estimate two ingredients in a mixture. They are useful only when the assumptions fit the situation. No calculation can recover information that the measurements never contained.
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