Understanding Cramer's Rule for Equations

Nov 18, 2024

Lecture Notes on Cramer's Rule

Introduction

  • Cramer's Rule: Method to solve a system of equations using determinants.
  • Applicable for systems with the same number of equations as unknowns.

System of Two Equations

  • General Form:
    • Equations: (ax + by = c) and (dx + ey = f)
  • Determinants:
    • Denominator: Formed by coefficients of x and y terms.
    • Numerator for x: Replace x coefficients with constants.
    • Numerator for y: Replace y coefficients with constants.

Example

  • Given equations, setup:
    • Denominator (both x and y):
      • [ \begin{bmatrix} 1 & -1 \ 1 & 4 \end{bmatrix} ]
    • Numerator for x:
      • [ \begin{bmatrix} -3 & -1 \ 17 & 4 \end{bmatrix} ]
    • Numerator for y:
      • [ \begin{bmatrix} 1 & -3 \ 1 & 17 \end{bmatrix} ]
  • Solution:
    • x = 1, y = 4

System of Three Equations

  • Maintain pattern from two-equation system.
  • Determinants:
    • Denominator: Formed by coefficients of x, y, and z.
    • Numerators: Replace coefficients of the variable being solved with constants.

Example

  • Setup for equations:
    • Denominator:
      • [ \begin{bmatrix} 2 & 3 & 1 \ -1 & 2 & 3 \ -3 & -3 & 1 \end{bmatrix} ]
    • Numerator for x:
      • [ \begin{bmatrix} 2 & 3 & 1 \ -1 & 2 & 3 \ 0 & -3 & 1 \end{bmatrix} ]
    • Numerator for y:
      • [ \begin{bmatrix} 2 & 3 & 1 \ -1 & -1 & 3 \ 0 & -3 & 1 \end{bmatrix} ]
    • Numerator for z:
      • [ \begin{bmatrix} 2 & 3 & 1 \ -1 & 2 & -1 \ 0 & -3 & 0 \end{bmatrix} ]
  • Solution:
    • Calculated using a graphing calculator:
      • x = 4, y = -3, z = 3

Conclusion

  • Cramer’s rule involves calculating determinants and is efficient with technology.
  • Relies on understanding of matrices and determinants.
  • Excellent for solving systems when technology is available.

Thank you for attending the lecture!