Step 1 :The system of equations is: \(3x + 2y - z = -5\), \(-x - y + z = 1\), and \(5x + 4y - 3z = -7\)
Step 2 :Add the first and second equations to eliminate z: \((3x + 2y - z) + (-x - y + z) = -5 + 1\) which simplifies to \(2x + y = -4\)
Step 3 :This gives us a new equation: \(2x + y = -4\)
Step 4 :Multiply the second equation by 3 and add it to the third equation to eliminate z: \(3(-x - y + z) + (5x + 4y - 3z) = 3*1 + -7\) which simplifies to \(2x + y = -4\)
Step 5 :This gives us the same equation again: \(2x + y = -4\)
Step 6 :Since we have two identical equations, the system of equations is dependent and has infinitely many solutions.
Step 7 :\(\boxed{\text{The solutions can be written in the form } (x, y, z) = (t, -2t-4, 3t+5) \text{ for any real number } t}\)