Lesson 16Lesson 16

Cramer's ruleCramer's rule

Cramer's ruleCramer's ruleCramer's rule is a method for solving systems Cramer's rule is a method for solving systems of linear equations using determinants.of linear equations using determinants.

The solution of the linear system:The solution of the linear system:

ax + by = eax + by = e

cx + dy = f arecx + dy = f are

x = e b y = a ex = e b y = a e

f df d c fc f

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

Coefficient matrixCoefficient matrix

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

a ba b

c dc d

Using Cramer's ruleUsing Cramer's ruleSolve 3x + 2y = -1Solve 3x + 2y = -1 4x - 3y = 104x - 3y = 10The coefficient matrix is 3 2The coefficient matrix is 3 2 4 -34 -3x = -1 2 y = 3 -1x = -1 2 y = 3 -1 10 -310 -3 4 104 10 3 2 3 23 2 3 2 4 -3 4 -34 -3 4 -3 x = x = 3-203-20 = = -17-17 =1 y = =1 y = 30+430+4 = =34 34 = -2= -2 -9-8 -17 -9-8 -17-9-8 -17 -9-8 -17 so solution is so solution is (1,-2)(1,-2)

Solve Solve

x + y = 1x + y = 1

x + 2y = 4x + 2y = 4

x = 1 1 y = 1 1x = 1 1 y = 1 1

4 24 2 1 41 4

1 1 1 11 1 1 1

1 2 1 21 2 1 2

x= x= 2-4 2-4 = = -2-2 = -2 y = = -2 y = 4 - 14 - 1= = 3 3 = 3 = 3

2-1 1 2-1 12-1 1 2-1 1

So solution is So solution is (-2,3)(-2,3)

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

Classifying systems by their Classifying systems by their solutionssolutions

1) if 1) if D isD is not equal to 0, the system has 1 not equal to 0, the system has 1 unique solution. ( unique solution. ( consistentconsistent))

2) if 2) if D = 0D = 0, but , but neither numerator is 0neither numerator is 0, the , the solution has solution has no solutions (no solutions (inconsistentinconsistent))

3) if 3) if D = 0D = 0 and and at least one of the at least one of the numerators is 0,numerators is 0, the system has an the system has an infinite infinite number of solutionsnumber of solutions ( (dependent and dependent and consistent)consistent)

Interpreting a denominator of 0Interpreting a denominator of 0

3x + 2y = 53x + 2y = 5 3x + 2y = 83x + 2y = 8 x = 5 2 x = 5 2 10-1610-16= = -6-6 y = 3 5 y = 3 5 24-1524-15==99 8 28 2 6-6 0 6-6 0 3 83 8 6-6 0 6-6 0 3 2 3 23 2 3 2 3 2 3 23 2 3 2

Division by zero is undefined, so Cramer's rule Division by zero is undefined, so Cramer's rule did not provide a solution. Neither of the did not provide a solution. Neither of the numerator's is zero, so there is no solutionnumerator's is zero, so there is no solution

solvesolve

3x + 2y = 53x + 2y = 5

6x + 4y = 106x + 4y = 10

x = 5 2 =x = 5 2 =20-2020-20 = = 00 y = 3 5 = y = 3 5 = 30-30 30-30 ==00

10 410 4 12-12 =0 12-12 =0 6 106 10 12-12 =0 12-12 =0

3 2 3 23 2 3 2

6 4 6 46 4 6 4

The denominators are 0 and both numerators are 0, The denominators are 0 and both numerators are 0, so there is an infinite number of solutions to the so there is an infinite number of solutions to the systemsystem

Use Cramer's rule to solveUse Cramer's rule to solve

2x + y = 62x + y = 6

6x + 3y = 186x + 3y = 18

2x + 4y = 122x + 4y = 12

x + 2y = -2x + 2y = -2