Question: If the line passing through (3,-4) and (-2,a) is parallel to the line given by the equation y+2x+3=0, find the value of a.

Solution:
Given,

Two points of a line are: (3,-4) and (-2,a)

Let the coordinates of one point (3,-4) be ($x_1,y_1$) and the coordinates of the other (-2,a) be ($x_2,y_2$), respectively.

So, slope of the line is given by ($m_1$) = $\dfrac{y_2 - y_1}{x_2 - x_1}$
$= \dfrac{a - (-4)}{-2 -3}$
$= \dfrac{a +4}{-5}$
$= - \dfrac{a+4}{5}$

Also,

Equation of the other line is y+2x +3 = 0
Slope of the line is ($m_2$) = $\dfrac{- coefficient \;of \;x}{coefficient\; of\; y}$
$= \dfrac{- 2}{1}$
$= - 2$

It is said that the lines are parallel. So, the slopes of these lines are equal.

$or, m_1 = m_2$
$or, - \dfrac{a +4}{5} = -2$
$or, \dfrac{a+4}{5} = 2$
$or, a + 4 = 5×2$
$or, a +4 = 10$
$or, a = 10-4$
$\therefore a = 6$

Therefore, the required value of a is 6.