Please I beg of you anyone help me with this system of equation question please

For what values of d and e will the three equations have...?
: x + 3y -2z=8
2x + y - 3z= 5
7x -4y + dz= e

if a) the system of equation have no solution
b) infinitely many solutions
c) a unique solution


* can anyone do this question using Gaussian elimination

Gaussian elimination is the ideal way to approach this problem. You can do it in a matrix or not. But a matrix is more convenient.

[1 3 -2 | 8]
[2 1 -3 | 5]
[7 -4 d | e]

Perform -2R1 + R2--> R2 and -7R1 + R3 --> R3

[1 3 -2 | 8]
[0 -5 1 | -11]
[0 -25 14+d | -56 + e]

Perform -5R2 + R3 --> R3

[1 3 -2 | 8]
[0 -5 1 | -11]
[0 0 9 + d | -1 + e]

Okay, if the last row has the form

0 0 0 | not zero

the system has no solution. This happens if d = -9 and e ≠ 1.

If the last row is

0 0 0 | 0

the system has infinitely many solutions. This happens if d = -9 and e = 1.

If the last row is

0 0 (not zero) | anything

the system has exactly one solution. This happens if d ≠ -9.