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)

A system of two linear equations in two variables:

has infinitely many (multiple) solutions if and only if the coefficients and constants are proportional:

Substituting the given coefficients:

From the first equality:

From the second equality:

• If $a = 4$, then $b = \frac{24}{4} = 6$, giving $a \cdot b = 4 \times 6 = 24$.

• If $a = -4$, then $b = \frac{24}{-4} = -6$, giving $a \cdot b = (-4) \times (-6) = 24$.

In both cases, the product of $a$ and $b$ is 24.