Combining like terms

The letter x stands for an unknown amount, and 5x means five of those amounts. The plain numbers, 3 and 4, are just counts with no unknown attached. An expression is a collection of these items, and items can only be combined with items of the same kind. Think of it as a crate full of boxes marked x and loose counters, where only matching kinds stack together.

Lena's mistake was treating every term as the same kind. She paired 5x with 3 to make 8x, and 2x with 4 to make 6x, then wrote 14x. That is like saying 5 apples plus 3 bananas makes 8 apples; the banana never turns into an apple. The two kinds stay separate all the way through the expression. Keep the kinds apart as you read, and the simplification becomes a sorting game.

The fix is grouping by kind. The x-terms, 5x and 2x, give 7x. The plain numbers, 3 and 4, give 7. Together the simplified expression is 7x + 7. The plus stays, because the two groups are different kinds and are not merged. Group first, then combine each group, and the plus is the seam between the kinds. Write each group's total in its own slot, and the seam is impossible to forget.

Check by choosing a value: if x = 1, the original is 5 + 3 + 2 + 4 = 14, and 7x + 7 = 7 + 7 = 14. Matching at a test value is a strong sign the combining is right. Try another value like x = 2, where the original gives 10 + 3 + 4 + 4 = 21 and the simplified form gives 14 + 7 = 21.