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:
7x +11y + αz = 13
5x + 4y + 7z = β
175x + 194y + 57z = 361
has infinitely many solutions, then α + β + 2 is equal to:
- 3
- 4
- 6
- 5
The Correct Option is B
Solution and Explanation
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:
- \(\alpha = -9\)
- \(\beta = 11\)
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