The solution is (x, y, z) = ((7-2z)/3, (z-2)/3, z), where z can be any real number.
Given system of linear equations
x - y + z = 3
3x + 2z = 7
x - 4y + 2z = 5
Analyze the reduced matrix
The reduced matrix shows that the system of equations is already in triangular form, which means it's ready for back substitution. We can solve for the variables one at a time, starting from the bottom row and moving up.
Solve for z
The bottom row of the matrix shows that z = z.
Therefore, z can be any real number.
Solve for y
The second row of the matrix shows that y = (z - 2)/3.
Substitute the expression for z from into this equation to get:
y = ((z - 2)/3) = (z - 2)/3.
Solve for x
The top row of the matrix shows that x = (7 - 2z)/3.
Substitute the expression for z into this equation to get:
x = (7 - 2(z))/3 = (7-2z)/3.
Therefore, the solution of the system of equations is (x, y, z) = ((7-2z)/3, (z-2)/3, z).