GATE 2026 CS (CS2) – Question 31
Consider the system of linear equations
$$ax+y=b$$
$$16x+ay=24$$
Suppose the values of $a$ and $b$ are chosen such that the system produces multiple solutions. Then the product $ab$ is __________. (answer in integer)
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Correct answer: 24
Explanation
For a $2\times2$ linear system to have multiple solutions, the two equations must be dependent. Thus their coefficients and constants must be proportional: $$\frac{a}{16}=\frac{1}{a}=\frac{b}{24}.$$ From $\frac{a}{16}=\frac{1}{a}$, we get $a^2=16$, so $a=\pm4$. Then $\frac{b}{24}=\frac{1}{a}$ gives $b=\pm6$ with the same sign as $a$. Therefore, $$ab=(\pm4)(\pm6)=24.$$ Hence the answer is 24.