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Simultaneous linear equations — KCSE Mathematics

KCSE Mathematics · 85 practice questions · 4 syllabus objectives · 4 revision lessons

29 easy39 medium17 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 a system of two simultaneous linear equations by elimination, substitution and graphical methods

Form and solve simultaneous equations from word problems (two unknowns, two conditions)

Determine whether a system of two linear equations has one solution, no solution or infinitely many solutions

Simultaneous linear equations

Revision Notes

Concise lesson notes for Simultaneous linear equations, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Solving Simultaneous Linear Equations

To solve a system of two simultaneous linear equations, we can use three methods: elimination, substitution, and graphical methods.

  1. Elimination Method: This involves adding or subtracting the equations to eliminate one variable.

    • Example: For the equations
      • 2x + 3y = 6
      • x - 2y = -4,
    • Multiply the second equation by 2 to align the coefficients:
      • 2x - 4y = -8.
    • Subtract the modified second equation from the first:
      • (2x + 3y) - (2x - 4y) = 6 - (-8)
      • This simplifies to 7y = 14, so y = 2.
    • Substitute y back into one of the original equations to find x.

  2. Substitution Method: Solve one equation for a variable, then substitute into the other.

    • Example: From x - 2y = -4, we can express x as x = 2y - 4, then substitute into the first equation.

  3. Graphical Method: Plot both equations on a graph and identify the intersection point, which represents the solution.

Each method can effectively find the solution to the simultaneous equations.

Key points to remember

  • Elimination involves adding or subtracting equations to remove a variable.
  • Substitution requires solving one equation for a variable first.
  • Graphical method shows the solution as the intersection point of lines.
  • All three methods yield the same solution for consistent equations.
  • Choose the method based on the equations' complexity and your preference.

Worked example

Solve the equations: 3x + 2y = 12 and 6x - 4y = 0 using elimination.

  • Multiply the first equation by 2: 6x + 4y = 24.
  • Subtract the second equation: 6x + 4y - (6x - 4y) = 24 - 0.
  • This simplifies to 8y = 24, so y = 3.
  • Substitute y back into the first equation: 3x + 2(3) = 12, hence x = 2.

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More lessons in this topic

Lesson 2: Solving Simultaneous Linear Equations

Objective: Form and solve simultaneous equations from word problems (two unknowns, two conditions)

To solve simultaneous equations from word problems, first, identify the two unknowns and formulate equations based on the conditions given. Use substitution or elimination methods to find the values of the unknowns.

Steps to follow:

  1. Define the unknowns clearly.
  2. Translate the word problem into mathematical equations.
  3. Solve the equations using an appropriate method.
  4. Check your solution in the original equations.

Example:
A farmer has chickens and cows. The total number of animals is 30. The total number of legs is 100. Form and solve the equations.
Let x = number of chickens and y = number of cows.

  • Equation 1: x + y = 30
  • Equation 2: 2x + 4y = 100

Using substitution or elimination, we can solve these equations to find x and y.

  • Identify unknowns and define them clearly.
  • Translate conditions into equations accurately.
  • Use substitution or elimination to solve.
  • Verify solutions in the original problem.

A school has boys and girls. There are 50 students in total. If there are 120 legs, form and solve the equations.
Let x = number of boys, y = number of girls.

  • Equation 1: x + y = 50
  • Equation 2: 2x + 2y = 120
    Solve: x = 30, y = 20.
Lesson 3: Understanding Solutions of Linear Equations

Objective: Determine whether a system of two linear equations has one solution, no solution or infinitely many solutions

In mathematics, a system of two linear equations can have three possible outcomes regarding solutions: one solution, no solution, or infinitely many solutions.

  1. One solution occurs when the lines intersect at a single point. This means the equations represent different lines with different slopes.
  2. No solution arises when the lines are parallel, indicating the equations represent the same slope but different y-intercepts.
  3. Infinitely many solutions happen when the two equations represent the same line, meaning they have identical slopes and y-intercepts.

To determine the number of solutions, you can use the following methods:

  • Graphical method: Plot the equations and observe the intersection points.
  • Algebraic method: Solve the equations simultaneously and analyze the results.

For example, consider the equations:

  1. 2x + 3y = 6
  2. 4x + 6y = 12

These equations are multiples of each other, so they represent the same line. Thus, they have infinitely many solutions.

  • One solution means lines intersect at one point.
  • No solution means lines are parallel with different intercepts.
  • Infinitely many solutions mean lines are identical.
  • Use graphical or algebraic methods to determine solutions.
  • Analyze slopes and intercepts of the equations.

Determine the nature of solutions for: 2x + y = 4 and 4x + 2y = 8.

  • The second equation is a multiple of the first.
  • Therefore, the system has infinitely many solutions.
Lesson 4: Solving Simultaneous Linear Equations

Objective: Simultaneous linear equations

Simultaneous linear equations consist of two or more equations with the same variables. To find the solution, we need to determine the values of the variables that satisfy all equations simultaneously. The two common methods for solving these equations are the substitution method and the elimination method.

Substitution Method:

  1. Solve one equation for one variable.
  2. Substitute that expression into the other equation.
  3. Solve for the second variable.
  4. Substitute back to find the first variable.

Elimination Method:

  1. Align the equations.
  2. Multiply one or both equations to make coefficients of one variable the same.
  3. Add or subtract the equations to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute back to find the other variable.

Both methods will yield the same solution. Remember to check your solution by substituting back into the original equations.

  • Simultaneous equations have common variables across multiple equations.
  • Substitution involves solving one equation for a variable first.
  • Elimination requires adjusting equations to cancel out a variable.
  • Solutions should satisfy all original equations.
  • Always verify solutions by substituting back.

Solve the equations: 2x + 3y = 6 and x - y = 1.

  • From x - y = 1, express x as x = y + 1.
  • Substitute into 2(y + 1) + 3y = 6.
  • Solve to find y = 0, then x = 1.

Sample Questions

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1
easySHORT ANSWER3 marks

Given the simultaneous equations 2x + 3y = 12 and 4x - 6y = 24, state the nature of their solutions. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
The equations are equivalent, representing the same line (1 mk)
There are infinitely many solutions (1 mk)
Any point on the line is a solution (1 mk)
2
easySHORT ANSWER4 marks

Given the equations: x + 2y = 10 and 3x - y = 5, solve for x and y using the substitution method. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Isolate x in the first equation to express it in terms of y (1 mk)
Substitute the expression for x into the second equation (1 mk)
Solve the resulting equation for y (1 mk)
Substitute back to find x and state both values (1 mk)
3
easySHORT ANSWER4 marks

Solve the simultaneous equations: 3x + 4y = 18 and 2x - y = 1. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Multiply the second equation by 4 to align coefficients of y (1 mk)
Add the modified second equation to the first to eliminate y (1 mk)
Solve the resulting equation for x (1 mk)
Substitute back to find y and state both values (1 mk)
4

Given the equations 2x + 3y = 12 and 4x + 6y = 24, determine the nature of the solutions for this system. (3 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Simultaneous linear equations" cover?

Simultaneous linear equations covers Solve a system of two simultaneous linear equations by elimination, substitution and graphical methods; Form and solve simultaneous equations from word problems (two unknowns, two conditions); Determine whether a system of two linear equations has one solution, no solution or infinitely many solutions, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Simultaneous linear equations?

HighMarks has 85 Simultaneous linear equations 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 Simultaneous linear equations for the KCSE exam?

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Quadratic expressions and equations

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