If the lines 2x + 3y = 5 and kx – 6y = 7 are parallel, then the value of k is:

1. –4

2. $–\dfrac{1}{4}$

3. 4

4. –5

For two lines to be parallel, their slopes must be equal.

Line 1 : 2x + 3y = 5

First, convert the equation 2x + 3y = 5 into the slope-intercept form, y = mx + c, to find its slope, m.

$3y = -2x + 5 \\[1em] y = -\dfrac{2}{3}x + \dfrac{5}{3} \\[1em] m_1 = -\dfrac{2}{3}$

Line 2: kx - 6y = 7

First, convert the equation kx - 6y = 7 into the slope-intercept form, y = mx + c, to find its slope, m.

$-6y = -kx + 7 \\[1em] y = \dfrac{-k}{-6}x + \dfrac{7}{-6} = \dfrac{k}{6}x - \dfrac{7}{6} \\[1em] m_2 = \dfrac{k}{6}$

The slopes of parallel lines are equal:

$\Rightarrow m$1 = m2 \\[1em] \Rightarrow -\dfrac{2}{3} = \dfrac{k}{6} \\[1em] \Rightarrow k = -\dfrac{2 \times 6}{3} \\[1em] \Rightarrow k = -\dfrac{12}{3} \\[1em] \Rightarrow k = -4.

Hence, option 1 is the correct option.