Representing Other Algebraic Relationships

Not everything is proportional. Learn to read a pattern, name the rule, and write it as an expression you can trust.

Build a row of squares out of matchsticks. One square takes 4 sticks.

Two squares side by side take 7, not 8 — the second square borrows the stick it shares.

squares made of matchsticks 1 square 4 sticks2 squares 7 sticks3 squares 10 sticks 1 4 2 7 3 10 4 13 10 31 in out each new square costs 3 sticks, not 4 — it borrows one

Three squares take 10. Four take 13. Each new square costs three sticks, never four.

So 100 squares take 3 × 100 sticks, plus the one extra at the very start: 301.

You did not have to build it. The rule 3n + 1 built it for you.

That is what an algebraic expression is for. The pattern is in the picture; the letter just writes it down so it works for any n, including the ones nobody will ever draw.

A growing tile pattern with rule 3n plus 2 n = 15n = 28n = 311n = 414 step of 3 each time, plus a starting 2 → rule: 3n + 2

From a pattern to a rule

Suppose a pattern of tiles goes 5, 8, 11, 14, …

The step is +3 each time, so the rule contains 3n. Now find the adjustment: at n = 1 we need 5, and 3 × 1 = 3, so we add 2.

The rule is 3n + 2. Test it on another term: n = 4 gives 3(4) + 2 = 14. ✓

The two-part recipe

For an arithmetic pattern: rule = (step) × n + (the value before the first step). Find the step, then back up one term to find the constant.

Why 3n + 2 is not proportional

Check the origin: at n = 0 the rule gives 2, not 0. The graph is a straight line lifted two units up.

The +2 is a starting amount that was there before any growth happened — a flat fee, a first tile, a head start.

Combining like terms

4x + 7 + 2x - 3 becomes 6x + 4. You may combine the 4x and 2x because both count xs. You may combine 7 and -3 because both are plain numbers.

You may not combine 6x and 4 — they count different things. That is like adding 6 apples to 4 dollars.

Circle before you combine

Circle each term with the sign in front of it. 4x + 7 + 2x - 3 has terms +4x, +7, +2x, -3. Sorting them this way stops sign errors.

The distributive property

a(b + c) = ab + ac. The multiplier reaches every term inside.

3(2x - 5) = 6x - 15. Notice the -5 also gets multiplied; the sign travels with it.

Backwards, this is factoring: 12x + 18 = 6(2x + 3).

Other relationships worth knowing

Inverse: xy = k. Doubling x halves y — like workers and time to finish a job.

Doubling growth: y = 2n. Each step multiplies rather than adds. This grows shockingly fast: 210 = 1024.

Custom operations: a problem may define its own symbol, like a ★ b = 2a - b. Just follow the definition exactly: 5 ★ 3 = 2(5) - 3 = 7.

Worked examples

Find the rule

A pattern of dots goes 7, 12, 17, 22, … How many dots in the 50th figure?

  1. Step: 12 - 7 = 5 It grows by 5 each time, so the rule holds 5n.
  2. At n = 1: 5(1) = 5, but we need 7, so add 2. Find the constant by fitting the first term.
  3. Rule: 5n + 2. Check n = 3: 17 ✓ Always test on a term you did not use to build it.
  4. 5(50) + 2 = 252 Substitute n = 50.

252 dots.

Simplify carefully

Simplify 5(2x - 3) - 2(x + 4).

  1. Distribute the 5: 10x - 15 The 5 hits both terms.
  2. Distribute the -2: -2x - 8 The minus sign is part of what you distribute — that is the step people miss.
  3. Combine: 10x - 2x = 8x and -15 - 8 = -23 Like terms with like terms.

8x - 23

A custom operation

Define a ◇ b = a2 - 2b. Compute 4 ◇ (3 ◇ 1).

  1. Innermost first: 3 ◇ 1 = 32 - 2(1) = 7 Parentheses come first, same as always.
  2. Now 4 ◇ 7 = 42 - 2(7) Substitute the result back in.
  3. = 16 - 14 = 2 Follow the definition exactly.

2

Words to know

Expression
A math phrase with no equals sign, like 3n + 5.
Term
A piece of an expression separated by + or -.
Coefficient
The number multiplying a variable — the 3 in 3n.
Like terms
Terms with exactly the same variable part; only these can be combined.
Arithmetic sequence
A list that grows by adding the same amount each time.

Mistakes to watch for

Forgetting to distribute to the second term. 3(x + 4) is 3x + 12, because the bracket means three lots of the whole thing — and the whole thing includes the 4.

Dropping a negative when distributing. -2(x + 4) = -2x - 8, not -2x + 8: the -2 multiplies both terms, so the second sign changes too.

Combining unlike terms. 5x + 3 is not 8x — x is an unknown quantity and 3 is just three. You can only add terms whose variable parts are identical.

Reading a pattern from two terms only. 2, 4, … could be +2 each time or doubling each time. Always test your rule on a third term before trusting it.

Using a letter for an unknown seems obvious now, but it took centuries. The Greeks did algebra with line segments, drawing lengths instead of writing symbols.

Diophantus of Alexandria, around 250 AD, used a single abbreviation for the unknown — the first step towards a symbol.

The full modern habit, where any letter can stand for any quantity, only settled in the 1600s with Viète and Descartes.

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