The system of linear equations x + y - z = 4 2 x - y - 5 z = 5 4 x + y - 7 z = 13 has infinitely many solutions. If the general solution is parameterized as x = a z + b and y = c z + d , then the value of a 2 + b 2 + c 2 + d 2 is equal to

Step-by-step solution

Given the system of equations: x + y - z = 4           . . . ( 1 ) 2 x - y - 5 z = 5       . . . ( 2 ) Since the system has infinitely many solutions, the third equation is a linear combination of the first two and provides no new information. We can solve for x and y in terms of z using just the first two equations. Adding equation (1) and equation (2) to eliminate y : ( x + y - z ) + ( 2 x - y - 5 z ) = 4 + 5 3 x - 6 z = 9 Dividing by 3: x - 2 z = 3 ⇒ x = 2 z + 3 Comp