In Introduction to Algebra you used letters to stand for numbers and built expressions such as 3x + 4. When an expression is set equal to something, you get an equation: 3x + 4 = 19. An equation is a statement that two things are equal, and solving it means finding the value of the letter that makes the statement true. In this lesson you will turn word problems into equations and solve linear equations with one variable – equations where the letter is not squared or multiplied by itself. These are the first steps towards solving almost any problem in algebra.

Expressions and equations

• An expression has no equals sign: 3x + 4, 2(n − 1), y ÷ 5.
• An equation has an equals sign with something on each side: 3x + 4 = 19.
• The value that makes an equation true is its solution (or root). For 3x + 4 = 19 the solution is x = 5, because 3 × 5 + 4 = 19.

The balance method

Think of an equation as a pair of balanced scales. The left pan holds 3 bags with the same unknown number of marbles in each, plus 4 loose marbles; the right pan holds 19 loose marbles. The scales balance. If you take the same thing off both pans, or share both sides into equal groups, they still balance. So the one rule of solving equations is: whatever you do to one side, do to the other side.

To get the letter on its own, undo what has been done to it using inverse operations: subtraction undoes addition, addition undoes subtraction, division undoes multiplication, and multiplication undoes division. Undo the operations in the reverse order: deal with any number added or subtracted first, then with the multiplying number.

Equation	Do this to both sides	Result
x + 7 = 15	subtract 7	x = 8
5x = 35	divide by 5	x = 7
x4 = 6	multiply by 4	x = 24
3x − 4 = 11	add 4, then divide by 3	3x = 15, so x = 5
2(x + 3) = 18	divide by 2, then subtract 3	x + 3 = 9, so x = 6

Always check by putting your answer back into the original equation. For 3x − 4 = 11 with x = 5: 3 × 5 − 4 = 15 − 4 = 11 ✓.

Letters on both sides? Take away the smaller letter term from both sides first. For 5x + 2 = 3x + 12, subtract 3x: 2x + 2 = 12, then 2x = 10 and x = 5. And the answer does not have to be a whole number: 4x + 9 = 3 gives 4x = −6, so x = −112 (look back at Directed Numbers if negatives worry you).

Turning word problems into equations

1. Read the problem and decide what the unknown is. Choose a letter for it and say what it stands for: “Let n be the number of tickets.”
2. Translate the words into an equation. Useful phrases: “increased by” or “more than” means +; “decreased by” or “less than” means −; “times”, “twice” or “of” means ×; “shared equally” means ÷; “is”, “gives” or “makes” means =.
3. Solve the equation with the balance method.
4. Answer the question in words, with units, and check that it makes sense in the story.

Words	Equation
A number increased by 7 is 15.	n + 7 = 15
Three times a number, minus 4, is 11.	3n − 4 = 11
Half of a number is 9.	n2 = 9
When 20 is shared equally among p people, each gets 4.	20p = 4

Worked examples

Example 1: Ages

Question: Shania is 3 years older than her brother Ravi. Their ages add up to 27. How old is each of them?

1. Let Ravi be r years old. Then Shania is r + 3.
2. Equation: r + r + 3 = 27, so 2r + 3 = 27.
3. Subtract 3: 2r = 24. Divide by 2: r = 12.
4. Check: 12 + 15 = 27 ✓.

Answer: Ravi is 12 and Shania is 15.

Example 2: A trip to the cinema

Question: A group pays $155 for cinema tickets at $45 each plus $20 for popcorn. How many tickets did they buy?

1. Let t be the number of tickets: 45t + 20 = 155.
2. Subtract 20: 45t = 135.
3. Divide by 45: t = 3.

Answer: 3 tickets.

Example 3: A rectangle

Question: A rectangle is 4 cm longer than it is wide. Its perimeter is 36 cm. Find its length and width.

1. Let the width be x cm; the length is x + 4.
2. Perimeter: x + (x + 4) + x + (x + 4) = 4x + 8 = 36.
3. Subtract 8: 4x = 28, so x = 7.
4. Check: 2 × (7 + 11) = 36 ✓.

Answer: Width 7 cm, length 11 cm.

Example 4: Consecutive numbers

Question: Three consecutive whole numbers add up to 51. Find them.

1. Let the smallest be n; the others are n + 1 and n + 2.
2. n + n + 1 + n + 2 = 51, so 3n + 3 = 51.
3. 3n = 48, so n = 16.

Answer: 16, 17 and 18.

Example 5: Brackets

Question: Solve 2(x + 3) = 18 by expanding the brackets first.

1. Expand: 2x + 6 = 18.
2. Subtract 6: 2x = 12.
3. Divide by 2: x = 6.

Answer: x = 6 (the same as dividing by 2 first).

Common mistakes

• Doing something to one side only. If you subtract 4 on the left, subtract 4 on the right too.
• Undoing in the wrong order. In 3x − 4 = 11, add the 4 first, then divide by 3.
• Changing the sign wrongly. From x + 7 = 15 you subtract 7; you do not get x = 15 + 7.
• Not saying what the letter means in a word problem, then answering the wrong question.

Practice questions

Try these yourself first, then open each answer to check your working.

Quick check

Next: Consumer Arithmetic: Bills, Best Buys, Profit and Loss

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 3, Strand 1.6 Algebra, Topic 1.6.6 Algebraic Equations – Learning Outcomes 1.6.6.1 (translating word problems into algebraic equations) and 1.6.6.2 (solving linear equations with one variable).

Attributions to media used in this post:

Balance scale IMGP9755.jpg – Nikodem Nijaki – CC BY-SA 3.0 (via Wikimedia Commons)

Updates

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.