Lesson 16
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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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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 dc f

  • D D , where D is the determinant of the coefficient matrix


Coefficient matrix

  • This matrix is the coefficients of x and y in the given equations

  • a b

  • c d


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)


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)


undefined

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


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)


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


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


Use Cramer's rule to solve

  • 2x + y = 6

  • 6x + 3y = 18

    2x + 4y = 12

    x + 2y = -2


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