Question

If the solution of the system of simultaneous linear equations: $$ x + y - z = 6, $$ $$ 3x + 2y - z = 5, $$ $$ 2x - y - 2z + 3 = 0 $$ is \( x = \alpha, y = \beta, z = \gamma \), then \( \alpha + \beta = ?\) 

• A. \( -7 \)
• B. \( 2 \)
• C. \( 1 \)
• D. \( -2 \)

Hint:

For solving a system of linear equations, use substitution or elimination to express one variable in terms of others and simplify step by step.

Answer 1

The Correct Option is B

To solve the system of simultaneous linear equations, we have the following equations:

• \( x + y - z = 6 \) 
• \( 3x + 2y - z = 5 \)
• \( 2x - y - 2z + 3 = 0 \)

First, simplify the third equation:

\( 2x - y - 2z + 3 = 0 \)

\( \Rightarrow 2x - y - 2z = -3 \)

We can write the system of equations as:

• (1) \( x + y - z = 6 \)
• (2) \( 3x + 2y - z = 5 \)
• (3) \( 2x - y - 2z = -3 \)

 

To eliminate \( z \), subtract equation (1) from equation (2):

\( (3x + 2y - z) - (x + y - z) = 5 - 6 \)

\( 2x + y = -1 \)   ...(4)

Next, eliminate \( z \) between equations (1) and (3):

\( (2x - y - 2z) - 2(x + y - z) = -3 - 2(6) \)

\( 2x - y - 2z - 2x - 2y + 2z = -15 \)

\( -3y = -15 \)

\( y = 5 \)

Substitute \( y = 5 \) into equation (4):

\( 2x + 5 = -1 \)

\( 2x = -6 \)

\( x = -3 \)

Now substitute \( x = -3 \) and \( y = 5 \) into equation (1) to find \( z \):

\( -3 + 5 - z = 6 \)

\( 2 - z = 6 \)

\( z = -4 \)

Now we have \( x = -3 \), \( y = 5 \), \( z = -4 \). Hence, \( \alpha = -3 \), \( \beta = 5 \), and \( \gamma = -4 \).

Finally, \(\alpha + \beta = -3 + 5 = 2\).

Answer 2

We are given the system of equations: \[ x + y - z = 6, \] \[ 3x + 2y - z = 5, \] \[ 2x - y - 2z + 3 = 0. \] Step 1: Convert to standard form Rearrange the third equation: \[ 2x - y - 2z = -3. \] Thus, the system of equations is: \[ x + y - z = 6, \] \[ 3x + 2y - z = 5, \] \[ 2x - y - 2z = -3. \] Step 2: Solve for variables Using the first equation: \[ x + y = z + 6 \Rightarrow z = x + y - 6. \] Substituting \( z = x + y - 6 \) into the second equation: \[ 3x + 2y - (x + y - 6) = 5. \] Simplify: \[ 3x + 2y - x - y + 6 = 5. \] \[ 2x + y + 6 = 5 \Rightarrow 2x + y = -1. \] Substituting \( z = x + y - 6 \) into the third equation: \[ 2x - y - 2(x + y - 6) = -3. \] Expanding: \[ 2x - y - 2x - 2y + 12 = -3. \] \[ -3y + 12 = -3. \] \[ -3y = -15 \Rightarrow y = 5. \] Substituting \( y = 5 \) into \( 2x + y = -1 \): \[ 2x + 5 = -1. \] \[ 2x = -6 \Rightarrow x = -3. \] Using \( z = x + y - 6 \): \[ z = -3 + 5 - 6 = -4. \] Step 3: Compute \( \alpha + \beta \) \[ \alpha + \beta = x + y = -3 + 5 = 2. \] Thus, the correct answer is: \[ \boxed{2} \]