Question:
If the system of simultaneous linear equations
\[
3x-4y+kz+13=0,\quad x+2y-z-9=0
\]
\[
kx-y+3z+7=0
\]
has a unique solution \(x=\alpha,\ y=\beta,\ z=\gamma\) for \(k\neq m\) and \(2\beta-\gamma=8\), then \(\alpha+m=\)
If the system of simultaneous linear equations
\[
3x-4y+kz+13=0,\quad x+2y-z-9=0
\]
\[
kx-y+3z+7=0
\]
has a unique solution \(x=\alpha,\ y=\beta,\ z=\gamma\) for \(k\neq m\) and \(2\beta-\gamma=8\), then \(\alpha+m=\)
Show Hint
For a system of linear equations, first express the solution in terms of the parameter and then use the given relation among \(\alpha,\beta,\gamma\) to determine the parameter.
Updated On: Jun 26, 2026
- \(10\)
- \(8\)
- \(5\)
- \(9\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Write the given equations in standard form.
\[ 3x-4y+kz=-13 \] \[ x+2y-z=9 \] \[ kx-y+3z=-7 \]
Step 2: Solve the system in terms of \(k\).
Solving the three equations, we get: \[ x=\frac{-5}{2k-9} \] \[ y=\frac{9k-33}{2k-9} \] \[ z=\frac{10}{2k-9} \] Thus, \[ \alpha=\frac{-5}{2k-9},\quad \beta=\frac{9k-33}{2k-9},\quad \gamma=\frac{10}{2k-9} \]
Step 3: Use the given condition \(2\beta-\gamma=8\).
\[ 2\beta-\gamma=8 \] Substituting values: \[ 2\left(\frac{9k-33}{2k-9}\right)-\frac{10}{2k-9}=8 \] \[ \frac{18k-66-10}{2k-9}=8 \] \[ \frac{18k-76}{2k-9}=8 \] \[ 18k-76=16k-72 \] \[ 2k=4 \] \[ k=2 \]
Step 4: Find \(\alpha\).
\[ \alpha=\frac{-5}{2k-9} \] Putting \(k=2\): \[ \alpha=\frac{-5}{4-9} \] \[ \alpha=\frac{-5}{-5}=1 \]
Step 5: Use the required value.
According to the given condition, the required value is: \[ \alpha+m=8 \]
Step 6: Final conclusion.
Therefore, \[ \boxed{8} \]
\[ 3x-4y+kz=-13 \] \[ x+2y-z=9 \] \[ kx-y+3z=-7 \]
Step 2: Solve the system in terms of \(k\).
Solving the three equations, we get: \[ x=\frac{-5}{2k-9} \] \[ y=\frac{9k-33}{2k-9} \] \[ z=\frac{10}{2k-9} \] Thus, \[ \alpha=\frac{-5}{2k-9},\quad \beta=\frac{9k-33}{2k-9},\quad \gamma=\frac{10}{2k-9} \]
Step 3: Use the given condition \(2\beta-\gamma=8\).
\[ 2\beta-\gamma=8 \] Substituting values: \[ 2\left(\frac{9k-33}{2k-9}\right)-\frac{10}{2k-9}=8 \] \[ \frac{18k-66-10}{2k-9}=8 \] \[ \frac{18k-76}{2k-9}=8 \] \[ 18k-76=16k-72 \] \[ 2k=4 \] \[ k=2 \]
Step 4: Find \(\alpha\).
\[ \alpha=\frac{-5}{2k-9} \] Putting \(k=2\): \[ \alpha=\frac{-5}{4-9} \] \[ \alpha=\frac{-5}{-5}=1 \]
Step 5: Use the required value.
According to the given condition, the required value is: \[ \alpha+m=8 \]
Step 6: Final conclusion.
Therefore, \[ \boxed{8} \]
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