Linear Algebra

Step 1: Solve the system of equations. We are given:

(1) x - 2y + z = 34

(2) 2x + y + z = 102

(3) x + y - 3z = 17

We will solve this using substitution or elimination.
Step 2: Eliminate one variable. Let’s eliminate from equations (1) and (2). Subtract (1) from (2):

(2) - (1): (2x + y + z) - (x - 2y + z) = 102 - 34

x + 3y = 68 ... (4)

Now eliminate from equations (1) and (3): Multiply (1) by 3 and add to (3):

3(x - 2y + z) = 3 * 34 = 102 => 3x - 6y + 3z = 102

(3) + 3 * (1): (x + y - 3z) + (3x - 6y + 3z) = 17 + 102

4x - 5y = 119 ... (5)

Step 3: Solve equations (4) and (5): From (4): Substitute into (5):

4(68 - 3y) - 5y = 119

272 - 12y - 5y = 119

272 - 17y = 119 => 17y = 153 => y = 9

Now, substitute into (4):

x + 3(9) = 68 => x = 68 - 27 = 41

Now substitute , into (1):

41 - 2(9) + z = 34 => 41 - 18 + z = 34 => z = 11

Step 4: Compute

x + y + z = 41 + 9 + 11 = 61