Testing an equality and building an algebraic expression

An equality is a balance scale: one side on the left, one side on the right. It is true when both pans weigh the same. When a letter hides inside, as in 3x + 2 = 14, the scale may balance for some values of x and not for others. Your detective job: test a value and see whether the scale holds.

The method never changes. Replace the letter with the value you are testing, work out the left side, then the right side, each on its own. If the two results are equal, the equality is true for that value. Otherwise it is false. Watch out for the classic trap: 3x means 3 times x, so for x = 4, 3x equals 12, not 34!

The other superpower in this lesson is the reverse path: starting from a real situation and building an algebraic expression. A subscription costing 12 euros plus 2 euros per movie? For n movies you pay 12 + 2n. The letter stands for the number we do not know yet, and the expression becomes a calculating machine that works for every value.

To translate without slipping, go word by word: "triple of x" gives 3x, "increased by 5" adds + 5, "the sum of" calls for brackets if you multiply afterwards. Writing a good expression is like writing a sentence: every word matters, and so does the order.

These two moves, testing and translating, are the foundations of all algebra. Take the time to master them: everything that follows will build on them.