Difference between revisions of "Cramer's Rule"
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== General Solution for 2 Variables == | == General Solution for 2 Variables == | ||
− | Given a system of two equations with constants <math> | + | Given a system of two equations with constants <math>a, b, c, d, r, s</math> |
<cmath>\begin{eqnarray*} | <cmath>\begin{eqnarray*} | ||
− | + | ax + cy &=& r\ | |
− | + | bx + dy &=& s | |
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
Line 21: | Line 21: | ||
<math>x = \frac{\begin{vmatrix} | <math>x = \frac{\begin{vmatrix} | ||
− | + | r & c \ | |
− | + | s & d \end{vmatrix}} | |
{\begin{vmatrix} | {\begin{vmatrix} | ||
− | + | a & c \ | |
− | + | b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix} | |
− | + | a & r \ | |
− | + | b & s \end{vmatrix}} | |
{\begin{vmatrix} | {\begin{vmatrix} | ||
− | + | a & c \ | |
− | + | b & d \end{vmatrix}} = \frac{sa - rb}{ad - cb}</math> | |
== Example in 3 Variables == | == Example in 3 Variables == |
Revision as of 20:07, 19 May 2008
Cramer's Rule is a method of solving systems of equations using matrices.
General Form for n variables
Cramer's Rule employs the matrix determinant to solve a system of n linear equations in n variables.
We wish to solve the general linear system for the vector . Here, is the coefficient matrix, is a column vector.
Let be the matrix formed by replacing the jth column of with .
Then, Cramer's Rule states that the general solution is
General Solution for 2 Variables
Given a system of two equations with constants
Cramer's Rule states that and can be found through determinants according to the following:
Example in 3 Variables
Here, $A = \left(
Thus,
\[M_1 = \left( \begin{array}{ccc} 14 & 2 & 3 & 11 & 1 & 2 & 11 & 3 & 1 \end{array} \right) \qquad M_2 = \left( \begin{array}{ccc} 1 & 14 & 3 & 3 & 11 & 2 & 2 & 11 & 1 \end{array} \right) \qquad M_3 = \left( \begin{array}{ccc} 1 & 2 & 14 & 3 & 1 & 11 & 2 & 3 & 11 \end{array} \right)\] (Error compiling LaTeX. Unknown error_msg)
We calculate the determinants:
Finally, we solve the system: