Computer Sciences > GATE 2026 SET-2 > Engineering Mathematics
Consider the system of linear equations given below.
ax + y = b
16x + ay = 24
Suppose the values of a and b are chosen such that the system of linear equations produce multiple solutions. Then the product of a and b is _______.
ax + y = b
16x + ay = 24
Suppose the values of a and b are chosen such that the system of linear equations produce multiple solutions. Then the product of a and b is _______.
Correct : 24
For the system ax + y = b and 16x + ay = 24 to have multiple solutions, the two equations must be dependent — meaning one is a scalar multiple of the other. This requires the ratios of corresponding coefficients to be equal.
Setting the ratios equal: a/16 = 1/a = b/24
From a/16 = 1/a, cross multiplying gives a2 = 16, so a = 4.
From 1/a = b/24, substituting a = 4 gives 1/4 = b/24, so b = 6.
Verification: with a = 4 and b = 6, the first equation becomes 4x + y = 6 and the second becomes 16x + 4y = 24, which divides to 4x + y = 6 — identical equations confirming infinite solutions.
Product of a and b = 4 × 6 = 24 ✓
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