Exercise 3.6 Class 10 Math Solutions – Linear Systems, Cramer's Rule & Inversion, Punjab Board

Solved Exercise 3.6 for Class 10 Mathematics, Punjab Board. Complete step-by-step notes covering solutions of simultaneous linear equations using Matrix Inversion Method and Cramer's Rule, real-world word problems, and scientific applications in drug discovery, neuroscience, and climate science.

This page provides complete solved solutions for Exercise 3.6 from Unit 3: Matrices and Determinants, Class 10 Mathematics, Punjab Board (PCTB). It covers solving systems of simultaneous linear equations in two variables using both the Matrix Inversion Method (X = A⁻¹ B) and Cramer's Rule (x = det(Ax)/det(A), y = det(Ay)/det(A)), solving real-world word problems, and exploring matrices in scientific computing. All questions are worked out step by step for Matric Part 2 students.
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Exercise Overview

Exercise 3.6 serves as the culminating application exercise of Unit 3: Matrices and Determinants for Class 10 Punjab Board Mathematics. It synthesizes all preceding techniques – matrix formulation, determinant calculation, adjoint transposition, and matrix multiplication – into powerful computational algorithms for solving systems of simultaneous linear equations.

The exercise trains students in two essential solution techniques: the Matrix Inversion Method (converting a system into AX = B and evaluating X = A⁻¹B = (1/|A|) adj(A) B) and Cramer's Rule (computing x = |A_x|/|A| and y = |A_y|/|A| through column replacements). Beyond standard equations, students solve complex real-world word problems involving geometric dimensions, monetary amounts, and kinematic relations. The curriculum also demonstrates how matrix linear systems drive modern breakthroughs in scientific theories, including molecular modeling in drug discovery, synaptic graph matrices in neuroscience, and multi-variable differential grids in climate science.

In Punjab Board matric examinations (including Lahore, Faisalabad, Rawalpindi, Gujranwala, Multan, and other BISE boards), Exercise 3.6 is universally recognized as the single most critical section for compulsory Section C long questions. Examiners almost invariably include a 4-mark or 8-mark question requiring students to solve a system via Matrix Inversion Method or Cramer's Rule. Working through these step-by-step solutions ensures students master the full methodology and achieve maximum marks.

FAQs

Are these Exercise 3.6 solutions according to the Punjab Board syllabus?
Yes. These solutions follow the Punjab Curriculum and Textbook Board (PCTB) syllabus for Class 10 Mathematics, covering all Unit 3 Exercise 3.6 linear systems and Cramer's rule for the 2026-2027 session.
What is the Matrix Inversion Method for solving linear equations?
Given a linear system AX = B, where A is the coefficient matrix, X is the variable matrix [[x], [y]], and B is the constant matrix, the solution is obtained by multiplying both sides by A⁻¹, giving X = A⁻¹B = (1/|A|) adj(A) B.
What is Cramer's Rule for solving a system of two linear equations?
Cramer's Rule calculates variables directly using determinants: x = |A_x| / |A| and y = |A_y| / |A|, where A_x and A_y are formed by replacing the x and y columns of matrix A with the constant matrix B.
When does a simultaneous linear system have no solution using matrices?
If the determinant of the coefficient matrix is zero (|A| = 0), matrix A is singular, meaning A⁻¹ does not exist and Cramer's Rule yields division by zero; thus, the system cannot be solved by these methods.
Do Matrix Inversion Method and Cramer's Rule give the same answers?
Yes, both methods solve the identical linear system and will always yield the exact same values for x and y when |A| ≠ 0.
How are real-world word problems solved using matrices in Exercise 3.6?
Word problems are first translated into two simultaneous linear equations with two unknowns, converted into matrix form AX = B, and then solved using either the Matrix Inversion Method or Cramer's Rule.
How are matrices applied in modern scientific fields like drug discovery and climate science?
Matrices model complex multi-variable interactions in molecular docking for drug discovery, neural network connections in neuroscience, and large-scale atmospheric grids for climate modeling.
Can students read these Exercise 3.6 solutions online?
Yes, the complete solved PDF notes for Exercise 3.6 are embedded to read online directly on MaryamNotes.pk with no compulsory download.
Are these Exercise 3.6 solutions free for students?
Yes, all Exercise 3.6 solutions on MaryamNotes.pk are 100% free to read online for all Class 10 students across all Punjab boards.