Solving Simultaneous Equations Using Matrices for SS 3

Class: SS 3
Term: First Term
Week: 6
Age: 14 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Matrices and Determinants
Previous Lesson: Matrices as Linear Trans
Topic: MATRICES AND DETERMINANTS ii. solution of 3 simultaneous
Subject Matter: Solution of 2 simultaneous equation equation

Specific Objectives

By the end of the lesson, pupils/students should be able to:

Cognitive Domain

• Represent a system of two simultaneous linear equations in matrix form.
• Calculate the determinant of a 2×2 matrix.
• Find the inverse of a 2×2 matrix.
• Apply the matrix inverse method to solve two simultaneous linear equations.

Affective Domain

• Appreciate the efficiency of the matrix method in solving simultaneous equations.
• Demonstrate patience and precision when performing matrix calculations.

Psychomotor Domain

• Accurately perform matrix multiplication required for solving equations.
• Systematically present the steps involved in solving simultaneous equations using matrices.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools (A suitable textbook)
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing the representation and solution of 2 simultaneous equations using matrices.
• Whiteboard and markers.
• Calculators (optional, for checking solutions).

Rationale for the Lesson

This lesson is important as it introduces a powerful and systematic method for solving simultaneous linear equations, which is fundamental in various fields of science, engineering, and economics. Mastering the matrix method strengthens students’ algebraic skills and provides a deeper understanding of linear systems, preparing them for more advanced mathematical concepts.

Prerequisite/Previous Knowledge

Students should have prior knowledge of:

• Solving simultaneous equations using substitution or elimination methods.
• Basic matrix operations such as addition, subtraction, and multiplication of matrices.
• Understanding of identity matrices.

Lesson Content/Board Summary

Solving Simultaneous Equations Using Matrices

Representing Simultaneous Equations in Matrix Form

A system of two simultaneous linear equations can be written in the matrix form (AX = B), where:

1. (A) is the coefficient matrix.
2. (X) is the variable matrix.
3. (B) is the constant matrix.

For a system:

(ax + by = e)

(cx + dy = f)

The matrix form is:

( begin{pmatrix} a & b \ c & d end{pmatrix} begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} e \ f end{pmatrix} )

So, (A = begin{pmatrix} a & b \ c & d end{pmatrix}), (X = begin{pmatrix} x \ y end{pmatrix}), and (B = begin{pmatrix} e \ f end{pmatrix}).

Determinant of a 2×2 Matrix

For a 2×2 matrix (A = begin{pmatrix} a & b \ c & d end{pmatrix}), the determinant, denoted as (det(A)) or (|A|), is calculated as:

( det(A) = ad – bc )

Inverse of a 2×2 Matrix

The inverse of a 2×2 matrix (A = begin{pmatrix} a & b \ c & d end{pmatrix}), denoted as (A^{-1}), is given by the formula:

( A^{-1} = frac{1}{det(A)} begin{pmatrix} d & -b \ -c & a end{pmatrix} )

This formula is valid only if (det(A) neq 0).

Solving Simultaneous Equations Using the Matrix Inverse Method

Given the matrix equation (AX = B), to solve for (X), we multiply both sides by (A^{-1}) (the inverse of matrix (A)):

( A^{-1}AX = A^{-1}B )

Since (A^{-1}A = I) (the identity matrix) and (IX = X), we get:

( X = A^{-1}B )

This formula allows us to find the values of the variables (x) and (y).

Example 1

Question: Solve the following simultaneous equations using the matrix method:

( 2x + 3y = 7 )

( x – y = 1 )

Solution:

Step 1: Represent the equations in matrix form (AX = B).

( begin{pmatrix} 2 & 3 \ 1 & -1 end{pmatrix} begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 7 \ 1 end{pmatrix} )

Here, (A = begin{pmatrix} 2 & 3 \ 1 & -1 end{pmatrix}), (X = begin{pmatrix} x \ y end{pmatrix}), (B = begin{pmatrix} 7 \ 1 end{pmatrix}).

Step 2: Calculate the determinant of matrix (A).

( det(A) = (2)(-1) – (3)(1) = -2 – 3 = -5 )

Step 3: Find the inverse of matrix (A).

( A^{-1} = frac{1}{-5} begin{pmatrix} -1 & -3 \ -1 & 2 end{pmatrix} = begin{pmatrix} 1/5 & 3/5 \ 1/5 & -2/5 end{pmatrix} )

Step 4: Use the formula (X = A^{-1}B) to find (X).

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 1/5 & 3/5 \ 1/5 & -2/5 end{pmatrix} begin{pmatrix} 7 \ 1 end{pmatrix} )

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} (1/5)(7) + (3/5)(1) \ (1/5)(7) + (-2/5)(1) end{pmatrix} )

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 7/5 + 3/5 \ 7/5 – 2/5 end{pmatrix} = begin{pmatrix} 10/5 \ 5/5 end{pmatrix} = begin{pmatrix} 2 \ 1 end{pmatrix} )

Answer: Therefore, (x = 2) and (y = 1).

Example 2

Question: Solve the following simultaneous equations using the matrix method:

( 3x – 2y = 8 )

( 4x + y = 7 )

Solution:

Step 1: Represent the equations in matrix form (AX = B).

( begin{pmatrix} 3 & -2 \ 4 & 1 end{pmatrix} begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 8 \ 7 end{pmatrix} )

Here, (A = begin{pmatrix} 3 & -2 \ 4 & 1 end{pmatrix}), (X = begin{pmatrix} x \ y end{pmatrix}), (B = begin{pmatrix} 8 \ 7 end{pmatrix}).

Step 2: Calculate the determinant of matrix (A).

( det(A) = (3)(1) – (-2)(4) = 3 – (-8) = 3 + 8 = 11 )

Step 3: Find the inverse of matrix (A).

( A^{-1} = frac{1}{11} begin{pmatrix} 1 & 2 \ -4 & 3 end{pmatrix} = begin{pmatrix} 1/11 & 2/11 \ -4/11 & 3/11 end{pmatrix} )

Step 4: Use the formula (X = A^{-1}B) to find (X).

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 1/11 & 2/11 \ -4/11 & 3/11 end{pmatrix} begin{pmatrix} 8 \ 7 end{pmatrix} )

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} (1/11)(8) + (2/11)(7) \ (-4/11)(8) + (3/11)(7) end{pmatrix} )

( begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 8/11 + 14/11 \ -32/11 + 21/11 end{pmatrix} = begin{pmatrix} 22/11 \ -11/11 end{pmatrix} = begin{pmatrix} 2 \ -1 end{pmatrix} )

Answer: Therefore, (x = 2) and (y = -1).

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Questioning

Teacher’s Activity: The teacher greets the students and asks them to recall methods for solving simultaneous linear equations they have learned previously (e.g., substitution, elimination). The teacher then introduces the matrix method as another powerful tool.

Students’ Activity: Students respond by mentioning substitution and elimination methods. They listen attentively to the introduction of the new method.

Learning Point: Prior methods recall

Step 2: Representing Equations in Matrix Form

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains how to represent a system of two simultaneous linear equations in the matrix form (AX = B), clearly identifying the coefficient matrix (A), the variable matrix (X), and the constant matrix (B). The teacher uses an example to demonstrate.

Students’ Activity: Students observe the demonstration and practice converting given equations into matrix form.

Learning Point: Matrix form representation

Step 3: Calculating the Determinant of a 2×2 Matrix

Time: 7 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher explains the formula for calculating the determinant of a 2×2 matrix, (det(A) = ad – bc). The teacher works through examples, ensuring students understand the process.

Students’ Activity: Students follow along, ask questions, and practice calculating determinants for various 2×2 matrices.

Learning Point: 2×2 determinant calculation

Step 4: Finding the Inverse of a 2×2 Matrix

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces the formula for the inverse of a 2×2 matrix, (A^{-1} = frac{1}{det(A)} begin{pmatrix} d & -b \ -c & a end{pmatrix}). The teacher emphasizes that the determinant must not be zero and demonstrates finding the inverse of a matrix used in a previous example.

Students’ Activity: Students pay attention to the formula and steps, taking notes and attempting to find inverses for simple matrices.

Learning Point: 2×2 matrix inverse

Step 5: Introducing the Solving Formula

Time: 3 minutes

Teaching Skill: Explanation

Teacher’s Activity: The teacher explains that once (A^{-1}) is found, the solution is obtained using the formula (X = A^{-1}B). The teacher briefly explains the derivation from (AX = B).

Students’ Activity: Students listen and note down the key formula (X = A^{-1}B).

Learning Point: Matrix solution formula

Step 6: Solving Simultaneous Equations (Worked Examples)

Time: 9 minutes

Teaching Skill: Problem Solving/Guided Practice

Teacher’s Activity: The teacher guides students through the complete process of solving a system of two simultaneous equations using the matrix method, following the steps outlined in the Board Summary (Example 1 and 2). The teacher encourages students to participate in each step.

Students’ Activity: Students actively participate in solving the examples, asking questions, and performing calculations.

Learning Point: Step-by-step equation solving

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Represent the system (3x – y = 5) and (x + 2y = 4) in matrix form.
2. Calculate the determinant of the coefficient matrix from question 1.
3. Find the inverse of the coefficient matrix from question 1.
4. State the formula used to solve for the variable matrix (X).

Students’ Activity: Pupils answer orally and in writing.

Learning Point: Matrix method application

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides students to copy the essential Board Summary notes, including the formulas and worked examples, into their notebooks.

Students’ Activity: Students copy the notes carefully into their notebooks.

Learning Point: Board summary recording

Step 9: Conclusion

Time: 1 minute

Teaching Skill: Consolidation

Teacher’s Activity: The teacher briefly summarizes the matrix method for solving two simultaneous equations, reiterating its importance and usefulness.

Students’ Activity: Students listen and confirm their understanding.

Learning Point: Matrix method consolidation

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following simultaneous equations using the matrix inverse method:

1. (5x + 2y = 12)
   (3x + y = 7)
2. (x – 4y = -10)
   (2x + 3y = 1)
3. (4x + 5y = 13)
   (x – 2y = -5)

Lesson Keywords

• Matrix – A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
• Determinant – A scalar value that can be computed from the elements of a square matrix.
• Inverse Matrix – A matrix that, when multiplied by the original matrix, yields the identity matrix.
• Simultaneous Equations – A set of equations containing multiple variables, where the values of the variables satisfy all equations simultaneously.
• Coefficient Matrix – A matrix containing the coefficients of the variables in a system of linear equations.

Differentiation

For students who struggle, provide additional guided practice with simpler coefficients and step-by-step prompts. For advanced learners, challenge them with equations that require rearrangement before forming the matrix or introduce systems with fractional coefficients.

Suggested Lesson Videos

Search on YouTube for: “Solving 2×2 simultaneous equations using matrices SS3”

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have charts prepared showing examples of simultaneous equations in matrix form, and perhaps a step-by-step solution for a simple system. Begin by reviewing students’ existing knowledge of simultaneous equations to build a bridge to the new method. Emphasize the systematic nature of the matrix method. During the lesson development, break down the process into clear, manageable steps: matrix representation, determinant calculation, inverse finding, and applying the solving formula. Encourage active participation during guided practice and be prepared to address common errors in matrix multiplication or determinant calculation. Students should copy the Board Summary notes after the main concepts and examples have been thoroughly discussed and understood to reinforce learning.