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

Options

A. A)`-28`
B. B)`-14`
C. C)`-31`
D. D)`-45`

Answer

To solve the given problem, we need to work with the three equations provided. Let's break it down step by step.

### Step 1: Write down the equations
We have the following three equations:
1. \( 3x + 4y - 2z + 9 = 17 \)
2. \( 7x + 2y + 11z + 8 = 23 \)
3. \( 5x + 9y + 6z - 4 = 18 \)

### Step 2: Simplify each equation
Let's simplify each equation by moving the constant terms to the right side.

1. From the first equation:
\[ 3x + 4y - 2z = 17 - 9 \implies 3x + 4y - 2z = 8 \]

2. From the second equation:
\[ 7x + 2y + 11z = 23 - 8 \implies 7x + 2y + 11z = 15 \]

3. From the third equation:
\[ 5x + 9y + 6z = 18 + 4 \implies 5x + 9y + 6z = 22 \]

Now we have the simplified equations:
1. \( 3x + 4y - 2z = 8 \) (Equation 1)
2. \( 7x + 2y + 11z = 15 \) (Equation 2)
3. \( 5x + 9y + 6z = 22 \) (Equation 3)

### Step 3: Add the equations
Now, we will add all three equations together:

\[ (3x + 4y - 2z) + (7x + 2y + 11z) + (5x + 9y + 6z) = 8 + 15 + 22 \]

### Step 4: Combine like terms
Combining the left side:
- For \(x\): \(3x + 7x + 5x = 15x\)
- For \(y\): \(4y + 2y + 9y = 15y\)
- For \(z\): \(-2z + 11z + 6z = 15z\)

Now, the left side becomes:
\[ 15x + 15y + 15z \]

For the right side:
\[ 8 + 15 + 22 = 45 \]

So, we have:
\[ 15x + 15y + 15z = 45 \]

### Step 5: Factor out the common term
Factoring out 15 from the left side:
\[ 15(x + y + z) = 45 \]

### Step 6: Solve for \(x + y + z\)
Dividing both sides by 15:
\[ x + y + z = \frac{45}{15} = 3 \]

### Step 7: Find \(x + y + z - 34\)
Now, we need to find the value of \(x + y + z - 34\):
\[ x + y + z - 34 = 3 - 34 = -31 \]

### Final Answer
The value of \(x + y + z - 34\) is \(-31\).

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