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

Step 1: For the system \(ax + y = b\) and \(16x + ay = 24\) to have infinitely many (multiple) solutions, the two equations must represent the same line, so their coefficients must be proportional: \(\frac{a}{16} = \frac{1}{a} = \frac{b}{24}\).
Step 2: From \(\frac{a}{16} = \frac{1}{a}\), cross multiplying gives \(a^2 = 16\), so \(a = 4\) or \(a = -4\).
Step 3: Case \(a = 4\): using \(\frac{1}{a} = \frac{b}{24}\), we get \(\frac{1}{4} = \frac{b}{24}\), so \(b = 6\). Product \(ab = 4 \times 6 = 24\).
Step 4: Case \(a = -4\): \(\frac{1}{-4} = \frac{b}{24}\), so \(b = -6\). Product \(ab = (-4) \times (-6) = 24\).
Step 5: In both valid cases the product \(ab\) equals the same value, confirming a unique answer regardless of sign choice.
Final Answer: \[\boxed{ab = 24}\]