Question

If the pair of equations ax + 2y = 7 and 3x + by = 16 represent parallel lines, then ab = ________.

Answer 1

If the system of equations $a_{1}x+b_{1}y+c_1=0\mathrm{and}{a}_{2}x+b_{2}y+c_2=0$ has no solution, i.e., they are parallel, then $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$.

Since, the system of equations ax + 2y = 7 and 3x + by = 16 represent parallel lines
Therefore,
$$\begin{gathered} \frac{a}{3}=\frac{2}{b}\ne \frac{-7}{-16} \\ \Rightarrow \frac{a}{3}=\frac{2}{b}\ne \frac{7}{16} \\ \Rightarrow \frac{a}{3}=\frac{2}{b} \\ \Rightarrow ab=6 \end{gathered}$$

Hence, If the pair of equations ax + 2y = 7 and 3x + by = 16 represent parallel lines, then ab = 6.