SOLUTION: Use cramer's rule or to determine that the system is inconsistent or contains dependent equations: 3=7y+1 2x=3y-1

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Question 152813: Use cramer's rule or to determine that the system is inconsistent or contains dependent equations:
3=7y+1
2x=3y-1
Answer by mducky2(62) About Me  (Show Source):
You can put this solution on YOUR website!
Cramer's rule means converting the system of equations into matrices. We need to find D, Dx, and Dy. Let's start by labeling the coefficients that we'll eventually put into matrices.
a1x + b1y = c1
a2x + b2y = c2

We need to make these equations below fit the ones above:

Equation 1:
3+=+7y+%2B+1
2+=+7y
7y+=+2
0x+%2B+7y+=+2
Because there are no x's in the equation, we can just assume the coefficient for x is 0. Therefore:
a%5B1%5D+=+0
b%5B1%5D+=+7
c%5B1%5D+=+2

Equation 2:
2x=3y-1
2x-3y=-1
Therefore:
a%5B2%5D+=+2
b%5B2%5D+=+-3
c%5B2%5D+=+-1


Now let's start by finding the determinant, D.
D+=+%28+matrix%28+2%2C+2%2C+a%5B1%5D%2C+b%5B1%5D%2C+a%5B2%5D%2C+b%5B2%5D+%29%29
D+=+%28+matrix%28+2%2C+2%2C+0%2C+7%2C+2%2C+-3+%29%29
D+=+%28a%5B1%5D%29%28b%5B2%5D%29+-+%28b%5B1%5D%29%28a%5B2%5D%29
D+=+%280%29%28-3%29+-+%287%29%282%29
D+=+0+-+14
D+=+-14

Now we can find the determinant Dx. This matrix is basically the same as D but with the first row replaced.
D%5Bx%5D+=+%28+matrix%28+2%2C+2%2C+c%5B1%5D%2C+b%5B1%5D%2C+c%5B2%5D%2C+b%5B2%5D+%29%29
D%5Bx%5D+=+%28+matrix%28+2%2C+2%2C+2%2C+7%2C+-1%2C+-3+%29%29
D%5Bx%5D+=+%28c%5B1%5D%29%28b%5B2%5D%29+-+%28b%5B1%5D%29%28c%5B2%5D%29
D%5Bx%5D+=+%282%29%28-3%29+-+%287%29%28-1%29
D%5Bx%5D+=+-6+-+%28-7%29
D%5Bx%5D+=+-6+%2B+7
D%5Bx%5D+=+1

Then, let's find the determinant Dy. This matrix is basically the same as D but with second row replaced.
D%5By%5D+=+%28+matrix%28+2%2C+2%2C+a%5B1%5D%2C+c%5B1%5D%2C+a%5B2%5D%2C+c%5B2%5D+%29%29
D%5By%5D+=+%28+matrix%28+2%2C+2%2C+0%2C+2%2C+2%2C+-1+%29%29
D%5By%5D+=+%28a%5B1%5D%29%28c%5B2%5D%29+-+%28c%5B1%5D%29%28a%5B2%5D%29
D%5By%5D+=+%280%29%28-1%29+-+%282%29%282%29
D%5By%5D+=+0+-+4
D%5By%5D+=+-4


We can finally solve for x and y.

x+=+D%5Bx%5D%2FD
x+=+1%2F-14
x+=+-1%2F14

y+=+D%5By%5D%2FD
y+=+-4%2F%28-14%29
y+=+2%2F7%29

The answer to the system is -1%2F14, 2%2F7


The system would be inconsistent if it had no solutions, but it obviously does. The system would be dependent if any variables remained in the solutions, but there are only numbers. Therefore, this system is neither inconsistent nor dependent.