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Here is the problem: John is 4 years older than Mary. Together they are 30. How old is Mary?

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Age problem with two unknown current ages (John, Mary) and two facts.

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Read the problem. Two people: John and Mary. Let a equals John prime s age now and b equals Mary prime s age now.

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Translate each fact into an equation. John minus Mary equals 4; John plus Mary equals 30.

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Write both equations in standard form. 1a minus 1b equals 4 and 1a plus 1b equals 30.

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Solve the system (elimination / Cramer prime s rule). Determinant equals 1 times 1 minus negative 1 times 1 equals 2. a equals (4 times 1 minus negative 1 times 30) / 2 equals 17, b equals (1 times 30 minus 4 times 1) / 2 equals 13.

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So the answer is John is 17 and Mary is 13 years old.

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Each sentence becomes one linear equation in the two current ages; two independent equations fix both ages.

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John minus Mary equals 4: 4 equals 4, which checks out.

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John plus Mary equals 30: 30 equals 30, which checks out.