Lesson 7: Expressions and One-Step Equations

A number machine at the fair takes your number, multiplies it by 3, and adds 5. You feed in 4 and out comes 17. Feed in 10 and out comes 35. Algebra is the language for describing machines like this: instead of trying numbers one at a time, we write 3x + 5 and let the letter x stand for any number. In this lesson you will evaluate expressions, translate English phrases into algebra, tidy expressions by combining like terms, and solve your first one-step equations — the gateway skill to all of algebra.

Expressions and variables

A variable is a letter that represents a number. An expression is a combination of numbers, variables and operations, such as 3x + 5 or 2(a − 3). Evaluating an expression means replacing each variable with a given number and working out the value, following the order of operations.

Worked example 1. Evaluate 3x + 5 when x = 4: 3 × 4 + 5 = 17 — exactly the fairground machine's output.

Worked example 2. Evaluate 2(a − 3) when a = 9: parentheses first, 2 × (9 − 3) = 2 × 6 = 12.

Worked example 3. Evaluate x² + 1 when x = 5: 5² + 1 = 25 + 1 = 26. Watch out: the exponent applies to x before you substitute anything else.

Translating words into algebra

Certain phrases always mean the same operation. "Seven more than a number" is n + 7. "A number decreased by 9" is n − 9. "The product of 6 and a number" is 6n (we write the number first, and no multiplication sign is needed). "Half of a number" is n/2. "Nine less than twice a number" is 2n − 9 — notice "less than" flips the order: you start from twice the number and subtract 9. That flip catches out more students than any other phrase, so slow down for it.

Combining like terms

Terms with the same variable part can be merged: 4x + 3x = 7x, just as 4 apples + 3 apples = 7 apples. Constants merge with constants: 2x + 5 + 6 = 2x + 11. But 4x + 3 stays as it is — x-terms and plain numbers are different species and cannot be combined, any more than apples and oranges make 7 applanges.

One-step equations

An equation states that two expressions are equal, and solving it means finding the value that makes it true. The golden rule: whatever you do to one side, do to the other, so the balance is kept. Use the inverse operation to free the variable.

Worked example 4. Solve x + 9 = 15. Subtract 9 from both sides: x = 6. Check: 6 + 9 = 15 ✓.

Worked example 5. Solve 5m = 35. Divide both sides by 5: m = 7. Check: 5 × 7 = 35 ✓.

Worked example 6. Solve k/3 = 8. Multiply both sides by 3: k = 24. And y − 4 = 11 needs 4 added to both sides: y = 15. Always finish by substituting your answer back into the original equation — the check takes ten seconds and catches nearly every slip.

In the real world: phone credit and puzzles

Maya had some phone credit, spent $13 on a game, and has $29 left. How much did she start with? Let m be the starting amount: m − 13 = 29. Add 13 to both sides: m = $42. Sudoku-style puzzles, "I'm thinking of a number" games, and working backwards from a final score in a video game (you ended with 240 points after doubling your score — you had 120) are all one-step equations in disguise.

Common mistakes

• Combining unlike terms. 3x + 4 is not 7x. Only terms with identical variable parts merge.
• Mishandling "less than". "9 less than a number" is n − 9, not 9 − n. The number you are starting from comes first.
• Doing the same operation to both sides — incorrectly. To solve x + 9 = 15 you subtract 9 (the inverse), not add 9 to both sides and get x = 24.
• Skipping the check. x = 6 in x + 9 = 15 checks out; if yours does not satisfy the original equation, the answer is wrong, however tidy it looks.

Exam Experience

1. Solve x − 7 = 13. A) 6 B) 7 C) 13 D) 20

2. What is the value of 4n − 2 when n = 6? A) 16 B) 20 C) 22 D) 26

3. Which expression means "9 less than twice a number n"? A) 9 − 2n B) 2n − 9 C) 2(n − 9) D) 9 − 2

4. (Constructed response) After spending $13 on a game, Maya has $29 of phone credit left. (a) Let m stand for the amount of credit Maya started with. Write an equation for this situation. (b) Solve your equation. (c) Check your answer by substituting it back into the equation, and explain in one sentence what the check proves.

Answer key and explanations

• 1. D — Add 7 to both sides: x = 13 + 7 = 20.
• 2. C — 4 × 6 − 2 = 24 − 2 = 22.
• 3. B — Twice the number is 2n; "9 less than" that is 2n − 9.
• 4. (a) m − 13 = 29. (b) m = 29 + 13 = 42. (c) 42 − 13 = 29 ✓ — the starting amount of $42, minus the $13 spent, gives exactly the $29 remaining, so the solution satisfies the original equation.

Scoring guide for question 4 (3 points)

• 1 point for the correct equation, m − 13 = 29 (or equivalent with the variable defined).
• 1 point for the solution m = 42, found by adding 13 to both sides.
• 1 point for an explicit check (42 − 13 = 29) with a sentence explaining that it confirms the answer fits the original situation.

Going further: expressions with teeth

Worked example 7. A rectangle has width 6 cm and length x cm. Its perimeter is P = 2x + 2 × 6 = 2x + 12. When x = 10, P = 20 + 12 = 32 cm. Geometry hands algebra its best problems: the expression describes every such rectangle at once, and substituting gives each particular one.

Worked example 8. Evaluate 5 − 2n when n = −1: 5 − 2 × (−1) = 5 − (−2) = 5 + 2 = 7. Substituting negative numbers is next year's everyday skill; the rule is to put the value in parentheses and let the sign rules work. Note how "minus a negative" quietly became "plus".

Worked example 9. Equations can have fraction and decimal answers. Solve 0.5x = 7: divide both sides by 0.5 (the same as doubling): x = 14. Solve x + 3/4 = 2: subtract 3/4 from both sides: x = 2 − 3/4 = 1 1/4. If an equation's answer is not whole, that is not a sign of error — check it the usual way: 1 1/4 + 3/4 = 2 ✓.

Worked example 10. A seedling 3 cm tall grows g centimetres and reaches 19 cm. Equation: 3 + g = 19, so g = 16 cm of growth. Tiny equation, but notice the shape: start + change = end. Half the word problems in mathematics are that shape wearing a costume.

Quick practice

• 1. Evaluate 7 − 3x when x = 2.
• 2. Solve x + 14 = 31.
• 3. Solve 9y = 108.
• 4. Translate: "a number divided by 5, then increased by 2".

Answers. 1. 7 − 6 = 1. 2. x = 17 (subtract 14 from both sides). 3. y = 12 (108 ÷ 9). 4. n/5 + 2 — division first in the phrase, and the +2 applies to the result.

Investigation: the number trick, exposed

A performer says: "Think of a number. Double it. Add 6. Halve the result. Subtract your original number. You are thinking of 3." Always 3 — and algebra shows why. Let the number be x. Double it: 2x. Add 6: 2x + 6. Halve: x + 3. Subtract the original: (x + 3) − x = 3, whatever x was. Test it with x = 10: 20, 26, 13, 3 ✓. The trick is not magic; it is an expression engineered so the variable cancels. You can build your own: any recipe that adds a constant and later removes the variable produces a "psychic" fixed answer. Now a quieter workout in the same spirit: a rectangle's perimeter is 34 cm and its width is 5 cm. Using P = 2(l + 5): 2(l + 5) = 34. Divide both sides by 2 (one step!): l + 5 = 17, so l = 12 cm. One-step equations often hide inside doubled brackets — divide the doubling away first and the familiar form appears.

Challenge corner

Three quick tests of everything in this lesson, back to back. First, evaluate 4(x − 2) + x when x = 7: the bracket gives 4 × 5 = 20, plus the final x gives 20 + 7 = 27. Substituting before simplifying is fine — but note that simplifying first (4x − 8 + x = 5x − 8) gives 5 × 7 − 8 = 27 by a shorter road. Second, solve 12 = x − 8: the equation arrives "backwards", with the variable on the right. No problem — equations can be read either way: add 8 to both sides and x = 20. Third, build one: a sister is 3 years older than her brother, so her age is b + 3 where b is his. If he is 9, she is 12; the expression, not the number, is the real answer, because it works for every age either of them will ever be.