Solving Systems of Equations Using Cramer's Rule

Introduction

• Cramer's Rule is a mathematical theorem used for solving systems of linear equations with as many equations as unknowns.
• In this lesson, we focus on solving a system of two linear equations with two variables.

Example Problem 1

System of Equations

1. (2x + 5y = 26)
2. (5x - 4y = -1)

Matrix Representation

• Represent the system in the form:
  • (A_1x + B_1y = C_1)
  • (A_2x + B_2y = C_2)
• Coefficients:
  • (A_1 = 2), (B_1 = 5), (C_1 = 26)
  • (A_2 = 5), (B_2 = -4), (C_2 = -1)

Determinant Calculation

• Determinant (D):
  • Formula: (A_1B_2 - B_1A_2)
  • Calculation: (2(-4) - 5(5) = -8 - 25 = -33)

Dx and Dy Calculation

• (D_x):

  • Replace (A_1) and (A_2) with (C_1) and (C_2)
  • Formula: (C_1B_2 - B_1C_2)
  • Calculation: (26(-4) - 5(-1) = -104 + 5 = -99)

• (D_y):

  • Replace (B_1) and (B_2) with (C_1) and (C_2)
  • Formula: (A_1C_2 - C_1A_2)
  • Calculation: (2(-1) - 5(26) = -2 - 130 = -132)

Solution

• (x): (\frac{D_x}{D} = \frac{-99}{-33} = 3)
• (y): (\frac{D_y}{D} = \frac{-132}{-33} = 4)
• Solution: ((x, y) = (3, 4))

Example Problem 2

System of Equations

1. (3x - 2y = -4)
2. (4x - y = 3)

Matrix Representation

• Coefficients:
  • (A_1 = 3), (B_1 = -2), (C_1 = -4)
  • (A_2 = 4), (B_2 = -1), (C_2 = 3)

Determinant Calculation

• Determinant (D):
  • Formula: (A_1B_2 - B_1A_2)
  • Calculation: (3(-1) - (-2)(4) = -3 + 8 = 5)

Dx and Dy Calculation

• (D_x):

  • Replace (A_1) and (A_2) with (C_1) and (C_2)
  • Formula: (C_1B_2 - B_1C_2)
  • Calculation: (-4(-1) - (-2)(3) = 4 + 6 = 10)

• (D_y):

  • Replace (B_1) and (B_2) with (C_1) and (C_2)
  • Formula: (A_1C_2 - C_1A_2)
  • Calculation: (3(3) - (-4)(4) = 9 + 16 = 25)

Solution

• (x): (\frac{D_x}{D} = \frac{10}{5} = 2)
• (y): (\frac{D_y}{D} = \frac{25}{5} = 5)
• Solution: ((x, y) = (2, 5))

Verification

• Plug solutions back into original equations to verify correctness.
  • Equation 1: (3(2) - 2(5) = -4)
  • Equation 2: (4(2) - 5 = 3)
• Both solutions satisfy the original equations.

Conclusion

• Cramer's Rule provides a systematic method for solving systems of linear equations.
• Important to verify solutions by substituting back into original equations.