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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

Lesson 16

Cramer's rule


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
Coefficient matrix

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

  • a b

  • c d


Using cramer s rule
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
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
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
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
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


Solve1
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
Use Cramer's rule to solve

  • 2x + y = 6

  • 6x + 3y = 18

    2x + 4y = 12

    x + 2y = -2


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