Question

If the system of linear equations
7x +11y + αz = 13
5x + 4y + 7z = β
175x + 194y + 57z = 361
has infinitely many solutions, then α + β + 2 is equal to:

• A. 3
• B. 4
• C. 6
• D. 5

Answer 1

The Correct Option is B

Step 1: Eliminate \(x\) and \(y\)

Multiply equation (i) by 10 and equation (ii) by 21, then subtract both from equation (iii):

\[ 10 \cdot (7x + 11y + \alpha z) + 21 \cdot (5x + 4y + 7z) - (175x + 194y + 57z) = 0 \]

Expand and simplify:

• \(10 \cdot 7x + 10 \cdot 11y + 10 \alpha z = 70x + 110y + 10\alpha z\)
• \(21 \cdot 5x + 21 \cdot 4y + 21 \cdot 7z = 105x + 84y + 147z\)
• \(175x + 194y + 57z = 175x + 194y + 57z\)

Subtracting, we get:

\[ z \cdot (10\alpha + 147 - 57) = 130 + 21\beta - 361 \]

Which simplifies to:

\[ z \cdot (10\alpha + 90) = 21\beta - 231 \]

Step 2: Solve for \(\alpha\)

Equating the coefficient of \(z\), we have:

\[ 10\alpha + 90 = 0 \]

Solving for \(\alpha\):

\[ \alpha = -9 \]

Step 3: Solve for \(\beta\)

Substitute \(\alpha = -9\) into the simplified equation:

\[ 130 - 361 + 21\beta = 0 \]

Which simplifies to:

\[ 21\beta = 231 \]

Solving for \(\beta\):

\[ \beta = 11 \]

Step 4: Verify the final condition

Check if the condition \(\alpha + \beta + 2 = 4\) is satisfied:

\[ -9 + 11 + 2 = 4 \]

The condition is satisfied.

Final Answer

The values are:

• \(\alpha = -9\)
• \(\beta = 11\)