Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations.
2x + y = 8
6x + 3y = 24
A. system is inconsistent
B. system contains dependent equations
b ?
Yes it's B, since it can be seen clearly that the simplified version of the second equation is equal to the first equation.
note the determinant of
2 1
6 3
is 6 - 6 = 0
therefore the solutions using Cramer's Rule are undefined
thank you guys :)
the solution would be:
( det 8 1 det 2 8
24 3 6 24
_____ ______
det 2 1 2 1
6 3 , 6 3 )
which is (0/0, 0/0) The zero in the denominator indicates that the system can not be solved using cramer's rule and is therefore not consistent and independent.
I have not read this but would conjecture that The zeros in the numerator indicate
that the system is consistent and dependent.
( If it were an inconsistent system, we would expect that both numerators would be non-zero, and not equal to each other)
To use Cramer's rule, we need to find the determinants of the coefficient matrix and augmented matrix of the system of equations.
Let's first write the system of equations in matrix form:
| 2 1 | | x | | 8 |
| 6 3 | * | y | = | 24 |
The determinant of the coefficient matrix is found by taking the determinant of the matrix:
D = | 2 1 |
| 6 3 |
D = (2 * 3) - (1 * 6)
D = 6 - 6
D = 0
Now let's find the determinant of the augmented matrix by replacing the coefficient matrix with the constant matrix on the right:
Dx = | 8 1 |
| 24 3 |
Dx = (8 * 3) - (1 * 24)
Dx = 24 - 24
Dx = 0
Dy = | 2 8 |
| 6 24 |
Dy = (2 * 24) - (8 * 6)
Dy = 48 - 48
Dy = 0
Now we can determine the nature of the system:
If D (the determinant of the coefficient matrix) is equal to 0, then the system is either inconsistent or contains dependent equations.
Since D = 0, we can conclude that the system contains dependent equations (option B).