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Standard XII

Mathematics

Consistency of Linear System of Equations

Find the valu...

Question

Find the values of $a$ and $b$ if the system of equations $a^{2} x - b y = a^{2} - b$ and $b x - b^{2} y = 2 + 4 b$ possess unique solution.

b \neq 0, a \neq 1

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b \neq 0, a \neq - 1

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b \neq 0, a = 1

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b = 0, a \neq 1

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Solution

The correct options are
A $b \neq 0, a \neq - 1$
C $b \neq 0, a \neq 1$
Here, $Δ = ∣ ∣ ∣ \begin{matrix} a^{2} & - b b & - b^{2} \end{matrix} ∣ ∣ ∣ = - a^{2} b^{2} + b^{2}$
We know for given equation to have unique solution $Δ \neq 0$
$\Rightarrow b^{2} (1 - a^{2}) \neq 0 \Rightarrow b \neq 0, a \neq \pm 1$

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Similar questions

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Find the values of a and b if the system of equations a2x−by=a2−b and bx−b2y=2+4b possess unique solution.

The correct options are A b≠0,a≠−1 C b≠0,a≠1Here, Δ=∣∣∣a2−bb−b2∣∣∣=−a2b2+b2 We know for given equation to have unique solution Δ≠0⇒b2(1−a2)≠0⇒b≠0,a≠±1

Find the values of $a$ and $b$ if the system of equations $a^{2} x - b y = a^{2} - b$ and $b x - b^{2} y = 2 + 4 b$ possess unique solution.