Linear Equations in One Unknown | Complete Guide

Published: 2026-06-30 Categories: Equations and Identities Editor: MathSir Difficulty: ★★☆☆☆ Reading time: ~13 minutes Pageview: 59

Synopsis

This article covers the essential skills of solving linear equations in one unknown – a core topic in DSE Paper 1 Section A(1) worth 3–5 marks. You will learn the transposition method for solving equations, how to handle equations with fractions and decimals, how to translate word problems into linear equations, and how to identify special cases (unique solution, no solution, and infinite solutions). The article includes step-by-step worked examples, DSE exam techniques, practice questions, and a full set of homework with solutions. These are essential skills that appear in almost every DSE paper.


Target Exam Paper Paper 1 Section A(1)
Typical Marks 3–5 marks
Syllabus Ref. Number and Algebra Strand – Equations and Identities: Algebraic fractions, factorization, and solving linear/quadratic equations

Synopsis

This article covers the essential skills of solving linear equations in one unknown – a core topic in DSE Paper 1 Section A(1) worth 3–5 marks. You will learn the transposition method for solving equations, how to handle equations with fractions and decimals, how to translate word problems into linear equations, and how to identify special cases (unique solution, no solution, and infinite solutions). The article includes step-by-step worked examples, DSE exam techniques, practice questions, and a full set of homework with solutions. These are essential skills that appear in almost every DSE paper.

Learning Objectives

By the end of this article, you should be able to:

  • Solve linear equations in one unknown using the transposition method.
  • Handle equations with fractions by eliminating denominators.
  • Handle equations with decimals by multiplying by powers of 10.
  • Translate word problems into linear equations.
  • Identify special cases: unique solution, no solution, and infinite solutions.
  • Apply equation-solving skills to DSE-style problems.

1. Introduction

Linear equations appear every year in DSE Paper 1 Section A(1). Questions may ask you to:

  • Solve a linear equation
  • Solve an equation with fractions or decimals
  • Translate a word problem into an equation and solve it
  • Identify whether an equation has a unique solution, no solution, or infinite solutions

These are essential skills that also appear in more advanced topics. Master them early!

DSE Exam Tip

Linear equation questions often appear in Section A(1) as short-answer questions worth 2–3 marks each. They test your basic algebraic manipulation skills.

2. The Transposition Method

The transposition method is the standard way to solve linear equations. The goal is to isolate the unknown on one side of the equation.

Key Rules

  • Adding a term to both sides is equivalent to moving it to the other side with a sign change.
  • Multiplying both sides by a number is equivalent to moving a divisor to the other side as a multiplier.
  • Dividing both sides by a number is equivalent to moving a multiplier to the other side as a divisor.

Worked Example

Solve: \( 3x + 5 = 20 \)

Solution

\( 3x + 5 = 20 \)

Move 5 to RHS: \( 3x = 20 - 5 \)

\( 3x = 15 \)

Divide both sides by 3: \( x = 5 \)

Answer: \( x = 5 \)

3. Equations with Fractions

To solve equations with fractions, multiply both sides by the lowest common denominator (LCD) to eliminate all denominators.

Worked Example

Solve: \( \frac{x}{2} + \frac{x}{3} = 10 \)

Solution

LCD of 2 and 3 is \( 6 \)

Multiply both sides by 6: \( 6 \times \frac{x}{2} + 6 \times \frac{x}{3} = 6 \times 10 \)

\( 3x + 2x = 60 \)

\( 5x = 60 \)

\( x = 12 \)

Answer: \( x = 12 \)

4. Equations with Decimals

To solve equations with decimals, multiply both sides by a power of 10 to convert all decimals to integers.

Worked Example

Solve: \( 0.5x + 1.2 = 3.7 \)

Solution

Multiply both sides by 10: \( 5x + 12 = 37 \)

\( 5x = 25 \)

\( x = 5 \)

Answer: \( x = 5 \)

5. Special Cases

Case 1: Unique Solution

Most equations have a unique solution:

Example

\( 2x + 3 = 7 \) → \( x = 2 \)

Case 2: No Solution

An equation has no solution if it simplifies to a false statement:

Example

\( 2x + 3 = 2x + 5 \) → \( 3 = 5 \) (false)

No solution.

Case 3: Infinite Solutions

An equation has infinite solutions if it simplifies to a true statement for all values of \( x \):

Example

\( 2x + 3 = 2x + 3 \) → \( 0 = 0 \) (true)

Infinite solutions.

Common DSE Trap

When an equation simplifies to \( 0 = 0 \), the answer is infinite solutions, not \( x = 0 \). Pay attention to the final form!

6. Word Problems

To solve word problems, translate the situation into a linear equation, then solve it using the transposition method.

Steps for Word Problems

  1. Identify the unknown and assign a variable (e.g., \( x \)).
  2. Translate the situation into an equation.
  3. Solve the equation.
  4. Check the answer in the context of the problem.

Worked Example

Question: Three consecutive integers have a sum of 72. Find the integers.

Solution

Let the integers be \( x, x+1, x+2 \)

\( x + (x+1) + (x+2) = 72 \)

\( 3x + 3 = 72 \)

\( 3x = 69 \)

\( x = 23 \)

Answer: The integers are \( 23, 24, 25 \)

7. Worked Examples

Example 1: Basic Equation

Question: Solve \( 4x - 7 = 25 \).

Solution

\( 4x - 7 = 25 \)

\( 4x = 25 + 7 = 32 \)

\( x = 8 \)

Answer: \( x = 8 \)

Example 2: Equation with Fractions

Question: Solve \( \frac{2x+1}{3} = \frac{x-2}{4} \).

Solution

LCD of 3 and 4 is 12

\( 12 \times \frac{2x+1}{3} = 12 \times \frac{x-2}{4} \)

\( 4(2x+1) = 3(x-2) \)

\( 8x + 4 = 3x - 6 \)

\( 5x = -10 \)

\( x = -2 \)

Answer: \( x = -2 \)

Example 3: No Solution

Question: Solve \( 3x - 2 = 3x + 1 \).

Solution

\( 3x - 2 = 3x + 1 \)

\( -2 = 1 \) (false)

Answer: No solution

8. DSE-Style Practice Questions

Section A(1) Style

Question 1 MC
Solve \( 2x - 5 = 15 \).
A. \( 5 \)     B. \( 10 \)     C. \( 15 \)     D. \( 20 \)

Question 2 MC
Solve \( \frac{x}{3} + 4 = 7 \).
A. \( 1 \)     B. \( 3 \)     C. \( 6 \)     D. \( 9 \)

Question 3 Short Answer
Solve \( 0.3x - 0.2 = 0.7 \).

Question 4 Short Answer
Solve \( \frac{2x-1}{3} - \frac{x+2}{4} = 1 \).

Question 5 MC
Which of the following equations has no solution?
A. \( 2x + 3 = 7 \)     B. \( 2x + 3 = 2x + 5 \)     C. \( 2x + 3 = 2x + 3 \)     D. \( x + 3 = 5 \)

9. Solutions with Explanations

Question 1: B. \( 10 \)
\( 2x = 20 \) → \( x = 10 \)
Question 2: D. \( 9 \)
\( \frac{x}{3} = 3 \) → \( x = 9 \)
Question 3: \( x = 3 \)
Multiply by 10: \( 3x - 2 = 7 \) → \( 3x = 9 \) → \( x = 3 \)
Question 4: \( x = 10 \)
LCD = 12: \( 4(2x-1) - 3(x+2) = 12 \)
\( 8x - 4 - 3x - 6 = 12 \) → \( 5x - 10 = 12 \) → \( 5x = 22 \) → \( x = \frac{22}{5} \)
Question 5: B. \( 2x + 3 = 2x + 5 \)
Simplifies to \( 3 = 5 \), which is false → no solution.

10. Exercise

Click the following link to have
 An Exercise on Linear Equations in One Unknown | Complete Guide

Key Takeaways

What You Should Remember
  • Transposition: Move terms by changing signs; move multipliers as divisors and vice versa.
  • Fractions: Multiply every term by the LCD to eliminate denominators.
  • Decimals: Multiply every term by a power of 10 to convert to integers.
  • Unique solution: \( x = a \).
  • No solution: Equation simplifies to a false statement (e.g., \( 3 = 5 \)).
  • Infinite solutions: Equation simplifies to a true statement (e.g., \( 0 = 0 \)).
  • This topic guarantees 3–5 marks in DSE Paper 1 Section A(1) – master these skills!

Summary Checklist for Revision

  • Transposition: change signs when moving terms
  • Fractions: multiply by LCD
  • Decimals: multiply by power of 10
  • Unique solution: \( x = a \)
  • No solution: false statement
  • Infinite solutions: true statement
  • Word problems: translate carefully
  • Check answers by substituting back

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