Question

The following is a system of linear equations

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

Equation (2):
\( 2x + y + z = 102 \)

Equation (3):
\( x + y - 3z = 17 \)

The value of \( x + y + z \) is ________. (rounded off to two decimal places)

Hint:

When solving systems of equations, reduce the system step-by-step using elimination or substitution. Plug values back to verify.

Answer 1

Correct Answer: 61

Step 1: Solve the system of equations.

We are given:
(1) \( x - 2y + z = 34 \)
(2) \( 2x + y + z = 102 \)
(3) \( x + y - 3z = 17 \)

Step 2: Eliminate one variable.

Eliminate \( z \) from equations (1) and (2) by subtracting (1) from (2):

\[ (2x + y + z) - (x - 2y + z) = 102 - 34 \]
\[ x + 3y = 68 \quad \text{(4)} \]

Now eliminate \( z \) from equations (1) and (3).
Multiply equation (1) by 3:

\[ 3x - 6y + 3z = 102 \]
Add this to equation (3):

\[ (x + y - 3z) + (3x - 6y + 3z) = 17 + 102 \]
\[ 4x - 5y = 119 \quad \text{(5)} \]

Step 3: Solve equations (4) and (5).

From equation (4):
\( x = 68 - 3y \)

Substitute into equation (5):

\[ 4(68 - 3y) - 5y = 119 \]
\[ 272 - 12y - 5y = 119 \]
\[ 272 - 17y = 119 \]
\[ 17y = 153 \Rightarrow y = 9 \]

Substitute \( y = 9 \) into equation (4):

\[ x + 3(9) = 68 \Rightarrow x = 41 \]

Substitute \( x = 41 \) and \( y = 9 \) into equation (1):

\[ 41 - 2(9) + z = 34 \]
\[ 41 - 18 + z = 34 \Rightarrow z = 11 \]

Step 4: Compute \( x + y + z \).

\[ x + y + z = 41 + 9 + 11 = 61 \]

Final Answer: 61.00