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:
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:
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!