Grade 11Maths Unit 4:4.6 Solutions of Systems of Linear Equations Using Cramer's Rule& Ex 4.11- 4.13

Introduction

This tutorial focuses on solving systems of linear equations using Cramer's Rule, based on the content from the Grade 11 Maths Unit 4.6 video by Saquama. Cramer's Rule is a mathematical theorem that provides an explicit formula for solving linear systems with as many equations as unknowns, which is particularly useful in mathematics and its applications.

Step 1: Understanding Cramer's Rule

• Definition: Cramer's Rule states that for a system of linear equations represented in matrix form (Ax = b), the solution for each variable can be found using determinants.
• Requirements:
  • The system must have the same number of equations as unknowns.
  • The determinant of the coefficient matrix (A) must be non-zero.

Step 2: Setting Up the Matrix

1. Identify the equations: Consider a system of equations:
   • For example:
     • (a_1x + b_1y = c_1)
     • (a_2x + b_2y = c_2)
2. Create the coefficient matrix (A):
   • Form matrix (A) from the coefficients of the variables.
   • Example: [ A = \begin{pmatrix} a_1 & b_1 \ a_2 & b_2 \end{pmatrix} ]
3. Create the constant matrix (B):
   • Form matrix (B) from the constants on the right-hand side of the equations.
   • Example: [ B = \begin{pmatrix} c_1 \ c_2 \end{pmatrix} ]

Step 3: Calculating the Determinant of Matrix A

• Determinant formula for a 2x2 matrix: [ \text{det}(A) = a_1b_2 - a_2b_1 ]
• This determinant must be non-zero to use Cramer’s Rule.

Step 4: Finding Variables Using Cramer’s Rule

1. Calculate (D): The determinant of the coefficient matrix (A).
2. Calculate (D_x):
   • Replace the first column of (A) with matrix (B) and calculate the determinant. [ D_x = \begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix} = c_1b_2 - c_2b_1 ]
3. Calculate (D_y):
   • Replace the second column of (A) with matrix (B) and calculate the determinant. [ D_y = \begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix} = a_1c_2 - a_2c_1 ]
4. Solve for variables:
   • Use the following formulas: [ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} ]

Step 5: Practice with Exercises

• Exercises 4.11 to 4.13: Apply Cramer’s Rule to the provided exercises in the video.
• Tips:
  • Double-check your determinant calculations to avoid mistakes.
  • Ensure that your matrix is set up correctly before making substitutions.

Conclusion

Cramer's Rule provides a systematic approach to solving systems of linear equations. By understanding how to set up matrices, calculate determinants, and apply the formulas for (x) and (y), you can solve various linear problems effectively. Practice the exercises to reinforce your understanding, and explore additional resources for further learning.