The condition for which the pair of equations $ax + 2y = 7$ and $3x + by = 16$ represent parallel lines is :

Step 1: Rewrite equations in standard form

First, we rewrite the given linear equations in the standard form $Ax + By + C = 0$. This helps in easily identifying the coefficients of $x$ and $y$, and the constant terms.

Step 2: Identify coefficients

From the rewritten equations, we can identify the coefficients for each variable and the constant terms. For the first equation, $a_1 = a$, $b_1 = 2$, and $c_1 = -7$. For the second equation, $a_2 = 3$, $b_2 = b$, and $c_2 = -16$.

Step 3: Apply condition for parallel lines

For two linear equations to represent parallel lines, the ratio of their $x$ coefficients must be equal to the ratio of their $y$ coefficients, but not equal to the ratio of their constant terms. This condition ensures that the lines have the same slope but different $y$ -intercepts.

Step 4: Substitute coefficients and solve

Substitute the identified coefficients into the condition for parallel lines. We get $\frac{a}{3} = \frac{2}{b}$ and $\frac{a}{3} \neq \frac{7}{16}$. From the first equality, we cross-multiply to find the relationship between $a$ and $b$.

Step 5: Derive the condition

From the equality $\frac{a}{3} = \frac{2}{b}$, cross-multiplying gives $a \times b = 3 \times 2$, which simplifies to $ab = 6$. This is the required condition for the lines to be parallel.

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