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Lesson 16 - PowerPoint PPT Presentation

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

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

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

• a b

• c d

• 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)

• 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)

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

• 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)

• 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

• 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

• 2x + y = 6

• 6x + 3y = 18

2x + 4y = 12

x + 2y = -2