Cramer's Rule Gabriel Cramer was a Swiss mathematician (1704- 1752) Reference from: : Fundamentals Methods of Mathematical Economics 4 th Edition (Page 103-107)
Introduction Cramer’s Rule is a method for solving linear simultaneous equations. It makes use of determinants and so a knowledge of these is necessary before proceeding. Cramer’s Rule relies on determinants
Coefficient Matrices You can use determinants to solve a system of linear equations. You use the coefficient matrix of the linear system. Linear System Coeff Matrix ax+by=e cx+dy=f d c b a
Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix ax+by=e cx+dy=f If detA 0, then the system has exactly one solution: and = ad – bc A d f b e x det A f c e a y det d c b a
Key Points The denominator consists of the coefficients of variables (x in the first column, and y in the second column). The numerator is the same as the denominator, with the constants replacing the coefficients of the variable for which you are solving.
Example - Applying Cramer’s Rule on a System of Two Equations Solve the system: 8x+5y= 2 2x-4y= -10 The coefficient matrix is: and So: and 4 2 5 8 42 ) 10 ( ) 32 ( 4 2 5 8 42 4 10 5 2 x 42 10 2 2 8 y
Solution: (-1,2) 1 42 42 42 ) 50 ( 8 42 4 10 5 2
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