Question

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 __________. (answer in
integer)

Hint:

For infinitely many solutions, the ratios of coefficients of x, y and the constant term across both equations must be equal: a/16 = 1/a = b/24.

Answer 1

Correct Answer: 24

This problem can also be solved using the determinant condition for a \(2\times 2\) system. For the system \(ax + y = b\), \(16x + ay = 24\) to have infinitely many solutions, the coefficient determinant must vanish: \(a \cdot a - 1 \cdot 16 = 0\), which gives \(a^2 - 16 = 0\), so \(a = \pm4\). Additionally, the two rows of the augmented matrix must be proportional, meaning the ratio of the second row to the first row is constant: \(16/a = a/1 = 24/b\). Taking \(a = 4\): the ratio is \(16/4 = 4\), so \(24/b = 4\) gives \(b = 6\), and \(ab = 24\). Taking \(a = -4\): the ratio is \(16/(-4) = -4\), so \(24/b = -4\) gives \(b = -6\), and \(ab = 24\) again. Either sign choice for \(a\) yields the same product because both \(a\) and \(b\) flip sign together. Hence the required value of \(ab\) is \(24\).