System of linear equations

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Question

If the system of linear equations,
$x + y + z = 6$
$x + 2 y + 3 z = 10$
$3 x + 2 y + λ z = μ$
has more than two solutions, then $μ - λ^{2}$ is equal to ______________

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Solution

$x + y + z = 6$ ___(1)
$x + 2 y + 3 z = 10$ _(2)
$3 x + 2 y + λ z = μ$ ___(3)
from (1) and (2)
If $z = 0 \Rightarrow x + y = 6$ and $x + 2 y = 10$ $\Rightarrow y = 4, x = 2$
$∴ (2, 4, 0) i s a s o l u t i o n o f t h e g i v e n s y s t e m$
          If $y = 0 \Rightarrow x + z = 6$ and $x + 3 z = 10$
         $\Rightarrow z = 2$ and $x = 4$
$\Rightarrow (4, 0, 2) i s a s o l u t i o n o f t h e g i v e n s y s t e m$
   Now equation (3) must satisfy the solutions $(2, 4, 0)$ and $(4, 0, 2)$ .
$\Rightarrow μ = 14, a n d 12 + 2 λ = μ \Rightarrow λ = 1$
$∴ μ - λ^{2} = 14 - 1$ = 13

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System of linear equations

Remember concepts with our Masterclasses.

If the system of linear equations,x+y+z=6x+2y+3z=103x+2y+λz=μhas more than two solutions, then μ-λ2 is equal to ______________

Ready to Test Your Skills?

free study material

If the system of linear equations,
$x + y + z = 6$
$x + 2 y + 3 z = 10$
$3 x + 2 y + λ z = μ$
has more than two solutions, then $μ - λ^{2}$ is equal to ______________