Lesson 16

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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
• 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