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.

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