If 3x + 4y – 2z + 9 = 17, 7x + 2y + 11z + 8 = 23 and 5x + 9y + 6z – 4 = 18, then what is the value of x + y + z – 34?

Introduction / Context:
This question involves solving a system of three linear equations in the three variables x, y, and z. Once the system is solved, we are asked to evaluate a simple linear expression in x, y, and z. Such questions test basic linear algebra skills and comfort with elimination or substitution methods.

Given Data / Assumptions:

• 3x + 4y - 2z + 9 = 17
• 7x + 2y + 11z + 8 = 23
• 5x + 9y + 6z - 4 = 18
• The required expression is x + y + z - 34.

Concept / Approach:
First, simplify each equation by moving constants to the right. This gives a cleaner system. Then use elimination or any standard method for solving simultaneous equations: remove one variable step by step until you find numeric values for x, y, and z. Finally, substitute those values into x + y + z - 34 to obtain the result. Working in an organized way helps avoid arithmetic mistakes.

Step-by-Step Solution:
Simplify the first equation: 3x + 4y - 2z + 9 = 17 gives 3x + 4y - 2z = 8. Simplify the second equation: 7x + 2y + 11z + 8 = 23 gives 7x + 2y + 11z = 15. Simplify the third equation: 5x + 9y + 6z - 4 = 18 gives 5x + 9y + 6z = 22. Now solve the system: (1) 3x + 4y - 2z = 8 (2) 7x + 2y + 11z = 15 (3) 5x + 9y + 6z = 22. Use elimination to remove variables. Solving this system yields x = 6/7, y = 34/21, and z = 11/21. Compute x + y + z = 6/7 + 34/21 + 11/21. Rewrite 6/7 as 18/21. Then x + y + z = 18/21 + 34/21 + 11/21 = 63/21 = 3. Now evaluate x + y + z - 34 = 3 - 34 = -31.

Verification / Alternative check:
We can substitute x = 6/7, y = 34/21, z = 11/21 back into each original equation to confirm that all three are satisfied. Plugging into the first gives 3*(6/7) + 4*(34/21) - 2*(11/21) + 9 = 17. Similar checks for the second and third equations confirm that these values solve the system. Therefore the derived value of x + y + z - 34 is reliable.

Why Other Options Are Wrong:
–28 and –14: These values would arise from incorrect arithmetic, such as mis adding fractions or mishandling the constant 34.
–22: This may result from a partially correct solution where one of x, y, or z is miscalculated.
–45: This is much too small and usually indicates a serious error in solving the system or forgetting to simplify the original equations correctly.

Common Pitfalls:
Learners sometimes forget to simplify the equations first, which makes elimination harder. Another common issue is making mistakes when handling fractions, especially when converting to a common denominator. Carefully aligning terms and checking each elimination step helps keep the work accurate and efficient.

Final Answer:
The value of x + y + z - 34 is -31.