What can you say about the numbers a, b, c and d?

[Gramsci]

Suppose a, b, c and d are constants such that a is not zero and the system below is consistent for all possible values of f and g. What can you say about the numbers a, b, c and d? Justify your answer.

Homework Equations

ax1+bx2=f
cx1+dx2=g

The Attempt at a Solution

I've been trying to think about it but I can't seem to get any closer to a solution. My basic thoughts is that we must perform some row reduction, but I'm not really sure how we should do it. I would love some help.
/gramsci

 

[Defennder]

You should row-reduce the 2nd row using the first row of the augmented matrix. Note that you have to multiply the first row by some multiple before adding it to the second row will yield a 0 for the first entry from the left of the second, which is why it is assumed that a is non-zero.

After you've done that, look at the contents of the 2nd row. Note that the system is inconsistent if the 2nd entry of the 2nd row is 0 and the 3rd entry of the 2nd row is non-zero. Write that out.

 

[Gramsci]

Ah, I multiply the first row by -c, which I guess is the right multiple and then add it to the second row which yields:
(-a, d-bc, g-fc) and after I restore the first row and add it to the second row I get the second row to be :
(0, d-c, g-c) and since c is 0 =
(0, d, g)
Then I understand what you meant. But it feels as if I performe some error during my calculations here, I'm guessing that I could've multiplied the first row wrong. If I did do any errors,please give me some leads.

/Thankfully, Gramsci