Question

If \( x, y, z \) are real numbers such that \( 4x + 2y + z = 31 \,\,\text{and}\,\, 2x + 4y - z = 19, \) then the value of \( 9x + 7y + z \) is 
 

• A. cannot be computed from the given information
• B. equals \( \frac{281}{3} \)
• C. equals \( \frac{182}{3} \)
• D. equals \( \frac{218}{3} \)

Hint:

When solving systems of linear equations, adding and subtracting equations is an effective way to eliminate variables.

Answer 1

The Correct Option is D

Step 1: Solve the system of equations.
We are given the system of equations: \[ 4x + 2y + z = 31 \text{(1)}, \] \[ 2x + 4y - z = 19 \text{(2)}. \]

Step 2: Add equations (1) and (2) to eliminate \( z \).
Adding equations (1) and (2) gives: \[ (4x + 2y + z) + (2x + 4y - z) = 31 + 19, \] \[ 6x + 6y = 50 $\Rightarrow$ x + y = \frac{50}{6} = \frac{25}{3}. \]

Step 3: Substitute \( y = \frac{25}{3} - x \) into equation (1).
Substitute \( y = \frac{25}{3} - x \) into equation (1): \[ 4x + 2\left( \frac{25}{3} - x \right) + z = 31, \] \[ 4x + \frac{50}{3} - 2x + z = 31, \] \[ 2x + z = 31 - \frac{50}{3} = \frac{93}{3} - \frac{50}{3} = \frac{43}{3}. \] Thus, \( z = \frac{43}{3} - 2x \).

Step 4: Final computation.
Substitute \( z = \frac{43}{3} - 2x \) into \( 9x + 7y + z \): \[ 9x + 7y + z = 9x + 7\left( \frac{25}{3} - x \right) + \left( \frac{43}{3} - 2x \right), \] \[ = 9x + \frac{175}{3} - 7x + \frac{43}{3} - 2x = 0 + \frac{218}{3}. \] Thus, the value of \( 9x + 7y + z \) is \( \frac{218}{3} \).

Step 5: Conclusion.
The correct answer is (D) equals \( \frac{218}{3} \).