
Simple matrix problem
Given the system:
3x + y + z + v = b1
x  y + z + v = b2
x + y  z + v = b3
x + y + z  v = b4
Fint the values for b1,b2,b3,b4 for which the system is consistent, in the form:
a1 * b1 + a2 * b2 + a3 * b3 + a4 * b4 = 0
and the answer is the row vector [a1,a2,a3,a4].
Solve the system with the values previously found, in the form:
x  b1 
y  b2 
z = C  b3  + t * d
v  b4 
Thanks!

Here's what I've got so far. The augmented matrix for the system is:
3 1 1 1 b1
1 1 1 1 b2
1 1 1 1 b3
1 1 1 1 b4
reduced eschelon form I made to be:
1 1/3 1/3 1/3 (1/3b1)
0 1 1/2 1/2 (3/4b2 + 1/4b1)
0 0 1 1 b3 (1/2b2 + 1/2b1)
0 0 0 0 (5/6b1 + 3/4b2 + 4/3b3 + 1/2b4)
Which means 5/6b1 + 3/4b2 + 4/3b3 + 1/2b4 has to equal 0 for the system to be consistent, and the final answer should be:
[5/6,3/4,4/3,1/2]
Is there a quick way to check the answer? Also, how do I solve the system with the values previously found, in the form:
x.....  b1 
y.....  b2 
z = C b3  + t * d
v...... b4 
Thanks!

Well, now I now that this reduced echelon form
1 1/3 1/3 1/3 (1/3b1)
0 1 1/2 1/2 (3/4b2 + 1/4b1)
0 0 1 1 b3 (1/2b2 + 1/2b1)
0 0 0 0 (5/6b1 + 3/4b2 + 4/3b3 + 1/2b4)
is wrong, but I don't know where I did something wrong... any help?