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Lesson 16

Lesson 16. Cramer's rule. Cramer's rule. Cramer's rule is a method for solving systems of linear equations using determinants. The solution of the linear system: ax + by = e cx + dy = f are x = e b y = a e f d c f

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Lesson 16

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  1. Lesson 16 Cramer's rule

  2. Cramer's rule • Cramer's rule is a method for solving systems of linear equations using determinants. • The solution of the linear system: • ax + by = e • cx + dy = f are • x = e b y = a e • f dc f • D D , where D is the determinant of the coefficient matrix

  3. Coefficient matrix • This matrix is the coefficients of x and y in the given equations • a b • c d

  4. Using Cramer's rule • Solve 3x + 2y = -1 • 4x - 3y = 10 • The coefficient matrix is 3 2 • 4 -3 • x = -1 2 y = 3 -1 • 10 -34 10 • 3 2 3 2 • 4 -3 4 -3 • x = 3-20 = -17 =1 y = 30+4 =34 = -2 • -9-8 -17 -9-8 -17 • so solution is (1,-2)

  5. Solve • x + y = 1 • x + 2y = 4 • x = 1 1 y = 1 1 • 4 21 4 • 1 1 1 1 • 1 2 1 2 • x= 2-4 = -2 = -2 y = 4 - 1= 3 = 3 • 2-1 1 2-1 1 • So solution is (-2,3)

  6. undefined • If the determinant of the coefficient matrix is 0, it makes the denominator of the solutions 0, which makes the solution undefined.

  7. Classifying systems by their solutions • 1) if D isnot equal to 0, the system has 1 unique solution. ( consistent) • 2) if D = 0, but neither numerator is 0, the solution has no solutions (inconsistent) • 3) if D = 0 and at least one of the numerators is 0, the system has an infinite number of solutions (dependent and consistent)

  8. Interpreting a denominator of 0 • 3x + 2y = 5 • 3x + 2y = 8 • x = 5 2 10-16= -6 y = 3 5 24-15=9 • 8 2 6-6 0 3 8 6-6 0 • 3 2 3 2 • 3 2 3 2 • Division by zero is undefined, so Cramer's rule did not provide a solution. Neither of the numerator's is zero, so there is no solution

  9. solve • 3x + 2y = 5 • 6x + 4y = 10 • x = 5 2 =20-20 = 0 y = 3 5 = 30-30 =0 • 10 4 12-12 =0 6 10 12-12 =0 • 3 2 3 2 • 6 4 6 4 • The denominators are 0 and both numerators are 0, so there is an infinite number of solutions to the system

  10. Use Cramer's rule to solve • 2x + y = 6 • 6x + 3y = 18 2x + 4y = 12 x + 2y = -2

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