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Linear equations (one variable) — KCSE Mathematics

KCSE Mathematics · 106 practice questions · 4 syllabus objectives · 4 revision lessons

36 easy37 medium33 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Solve linear equations in one variable including those with brackets and fractions

Form and solve linear equations from word problems involving age, money, distance-time and mixture situations

Solve linear equations involving absolute values: |ax + b| = c

Linear equations (one variable)

Revision Notes

Concise lesson notes for Linear equations (one variable), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Solving Linear Equations in One Variable

To solve linear equations in one variable, including those with brackets and fractions, follow these steps:

  1. Remove brackets by applying the distributive property. For example, for the equation 2(x + 3) = 12, distribute to get 2x + 6 = 12.
  2. Combine like terms on each side of the equation. This simplifies the equation.
  3. Isolate the variable by performing inverse operations (addition, subtraction, multiplication, or division) to get the variable on one side.
  4. Check your solution by substituting back into the original equation to ensure both sides are equal.

Example: Solve 3(x - 2) + 4 = 10.

  • Distribute: 3x - 6 + 4 = 10.
  • Combine like terms: 3x - 2 = 10.
  • Add 2 to both sides: 3x = 12.
  • Divide by 3: x = 4.

Final answer: x = 4. Always confirm by substituting back to check if the left side equals the right side.

Key points to remember

  • Use the distributive property to remove brackets.
  • Combine like terms to simplify the equation.
  • Isolate the variable using inverse operations.
  • Always check your solution by substitution.

Worked example

Solve the equation: 2(x + 5) = 20.

  • Distribute: 2x + 10 = 20.
  • Subtract 10: 2x = 10.
  • Divide by 2: x = 5. Final answer: x = 5.

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Lesson 2: Solving Linear Equations from Word Problems

Objective: Form and solve linear equations from word problems involving age, money, distance-time and mixture situations

In solving linear equations from word problems, it is essential to first identify the unknown variable and then translate the problem into a mathematical equation. Follow these steps:

  1. Read the problem carefully. Identify what is being asked.
  2. Define the variable. Let the unknown be represented by a letter (e.g., let x = age).
  3. Translate the words into an equation. Use mathematical operations based on the problem context.
  4. Solve the equation. Isolate the variable to find its value.
  5. Check your answer. Substitute back to ensure it fits the original problem.

Example: A mother is 4 years older than her son. If the son is x years old, write and solve an equation to find their ages if the mother is 28 years old.

Solution:

  1. Define the variable: Let x = son’s age.
  2. The mother’s age is x + 4.
  3. Set up the equation: x + 4 = 28.
  4. Solve: x = 28 - 4; x = 24.
  5. Conclusion: The son is 24 years old, and the mother is 28 years old.
  • Identify the unknown variable in the problem.
  • Translate the problem into a mathematical equation.
  • Isolate the variable to solve the equation.
  • Check your answer by substituting back into the problem.
  • Practice with different contexts like age or distance.

A person travels 60 km at a speed of x km/h. If the journey takes 3 hours, find x.

Solution:

  1. Set up the equation: 60 = 3x.
  2. Solve for x: x = 60/3; x = 20 km/h.
Lesson 3: Solving Linear Equations with Absolute Values

Objective: Solve linear equations involving absolute values: |ax + b| = c

To solve equations of the form |ax + b| = c, follow these steps:

  1. Identify cases: The absolute value equation can be split into two cases:
    • Case 1: ax + b = c
    • Case 2: ax + b = -c
  2. Solve each case: Solve for x in both cases.
  3. Check solutions: Substitute back into the original equation to ensure both solutions are valid.

Example: Solve |2x - 3| = 5.

  • Case 1: 2x - 3 = 5

    • 2x = 8
    • x = 4

  • Case 2: 2x - 3 = -5

    • 2x = -2
    • x = 1

Final solutions: x = 4 and x = 1. Both should be checked in the original equation, confirming they are valid.

  • Split the absolute equation into two cases.
  • Solve each case separately for x.
  • Check all solutions in the original equation.

Solve |3x + 2| = 7.

  • Case 1: 3x + 2 = 7 → 3x = 5 → x = 5/3.
  • Case 2: 3x + 2 = -7 → 3x = -9 → x = -3.
Lesson 4: Understanding Linear Equations in One Variable

Objective: Linear equations (one variable)

A linear equation in one variable is an equation that can be expressed in the form ax + b = 0, where a and b are constants and x is the variable. To solve for x, you need to isolate it on one side of the equation.

Steps to solve a linear equation:

  1. Move the constant term to the other side by subtracting or adding it.
  2. Divide both sides by the coefficient of x to isolate it.

Example: Solve the equation 3x + 6 = 0.

  • Subtract 6 from both sides: 3x = -6.
  • Divide both sides by 3: x = -2.

This means the solution to the equation is x = -2. Always remember to check your solution by substituting it back into the original equation to ensure both sides are equal.

  • Linear equations are in the form ax + b = 0.
  • To solve, isolate the variable x on one side.
  • Always check your solution by substituting back.

Solve the equation 2x - 4 = 0.

  • Add 4 to both sides: 2x = 4.
  • Divide by 2: x = 2.

Sample Questions

Read 3 questions and answers free. Sign up to access all 106 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

Identify the solutions to the equation |3x + 1| = 7. (4 marks)

Answer & marking scheme

Part (a) — 1 mark
3x + 1 = 7 or 3x + 1 = -7 (1 mk)
Part (b) — 3 marks
3x = 6, so x = 2 (1 mk)
3x = -8, so x = -/3 (1 mk)
Both values x = 2 and x = -/3 identified (1 mk)
2
easySHORT ANSWER2 marks

A shop sells notebooks at Ksh 50 each. If a customer buys n notebooks, the total cost is Ksh 50n. If the customer pays Ksh 300, how many notebooks can they buy? (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Set up the equation: 50n = 300. (1 mk)
Solve for n: n = 300 / 50 = 6. (1 mk)
3
easySHORT ANSWER2 marks

Determine the value of y in the equation: (y - 5)/3 + 2 = 4. [2 marks]

Answer & marking scheme

Part (a) — 2 marks
Subtract 2 from both sides to simplify to (y - 5)/3 = 2 (1 mk)
Multiply through by 3 to find y - 5 = 6 and solve to get y = 11 (1 mk)
4

Solve for x in the equation: 2(x + 3) = 4x - 8. [3 marks]

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Frequently asked questions

What does the KCSE Mathematics topic "Linear equations (one variable)" cover?

Linear equations (one variable) covers Solve linear equations in one variable including those with brackets and fractions; Form and solve linear equations from word problems involving age, money, distance-time and mixture situations; Solve linear equations involving absolute values: |ax + b| = c, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Linear equations (one variable)?

HighMarks has 106 Linear equations (one variable) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Linear equations (one variable) for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Linear equations (one variable)?

KNEC Aligned

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Related topics in KCSE Mathematics

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Algebraic expressions (expansion, factorization, simplification)
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Simultaneous linear equations

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