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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

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