Question

The value of \( a \), for which the following system of equations \[ 2x + y + 3z = a, x + z = 2, y + z = 2, \] is consistent, is

• A. 6
• B. 4
• C. 3
• D. 2

Hint:

When solving systems of equations, express variables in terms of others to simplify the process and check for consistency.

Answer 1

The Correct Option is A

We are given the system of linear equations: \[ 2x + y + 3z = a, x + z = 2, y + z = 2. \]

Step 1: Solve for \( x \), \( y \), and \( z \).
From the second equation, \( x + z = 2 \), we can express \( x = 2 - z \). Similarly, from the third equation, \( y + z = 2 \), we can express \( y = 2 - z \).

Step 2: Substitute into the first equation.
Substitute \( x = 2 - z \) and \( y = 2 - z \) into the first equation: \[ 2(2 - z) + (2 - z) + 3z = a. \] Simplifying this: \[ 4 - 2z + 2 - z + 3z = a $\Rightarrow$ 6 + 0z = a. \] Thus, \( a = 6 \).

Final Answer: \[ \boxed{\text{(A) } 6}. \]