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Mathematics Year 7: First-degree equations: recognising and translating

Mathematics, Year 7 (Portugal): An equation asks for which value of the unknown an equality is true, and writing a situation as an equation is the first step to solving it.

An equality with an unknown

An equation is an equality with an unknown, a letter that holds the place of a number you do not know yet. In 4x + 5 = 29 the term with the unknown is 4x, and the independent terms (no letter) are 5 and 29. The solution is the value of the unknown that makes the equality true. Test: x = 5 gives 4 × 5 + 5 = 25, not 29; x = 6 gives 4 × 6 + 5 = 29, true; x = 7 gives 33, false. So 6 is the solution. An equation is a question: for which value of x is this true?

One, none or many solutions

A statement like 3 + 4 = 7 just says something; an equation asks, and the answer depends on x. Like a balance, both sides must weigh the same. x − 2 = 0 is true only for x = 2: possible and determinate, one solution. x + 2 = x + 1 is never true, because the left side is always 1 more than the right: impossible, no solution. x + 2 = x + 2 is true for every number: possible and indeterminate. You can often tell which case it is by thinking, before any calculation.

From a story to an equation, and back

Four friends each buy the same notebook, costing x euros, and add a pen of 2.50 euros. They pay 14.50 euros in all. Equation: 4x + 2.5 = 14.5 (unknown term 4x; independent terms 2.5 and 14.5). Test 2, 3 and 4: 4 × 2 + 2.5 = 10.5; 4 × 3 + 2.5 = 14.5; 4 × 4 + 2.5 = 18.5. Only 3 works: each notebook costs 3 euros. Back the other way: 3x − 2 = 10 reads "three times a number, minus 2, gives 10", and x = 4 works: 3 × 4 − 2 = 10. And "a number plus 9 gives 30" is x + 9 = 30.

"Fewer than" is not "minus" in the order you hear it

Bia scored 6 points fewer than Nuno, who scored x. Many people write 6 − x, following the order of the words. It sounds right, so test it with a number: if Nuno scored 20, Bia scored 14, and 20 − 6 = 14. But 6 − 20 = −14, and nobody scores −14 points. The larger amount comes first: Bia's points are x − 6. So "Bia scored 14" becomes x − 6 = 14, and x = 20 fits. Whenever you translate a sentence, try a real number in your expression before trusting it.

Fixed fee plus a price per use

Equations appear whenever something is unknown and there is a condition to meet. Take renting a bike for a fixed 3 euros plus 2 euros per hour, when you paid 11 euros: how many hours? With x hours, the equation is 3 + 2x = 11, and x = 4 works: 3 + 2 × 4 = 11. The same shape fits a taxi fare, a phone plan, a tank being filled, a savings goal. Spreadsheets and programs solve equations all the time, but somebody first has to write the situation as an equation.

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