Show that the lines x + 2y – 5 = 0 and 2x + 4y + 9 = 0 are parallel.

Given,

⇒ x + 2y – 5 = 0

Converting x + 2y - 5 = 0 in the form y = mx + c we get,

⇒ 2y = -x + 5

⇒ y = $\dfrac{-x}{2} + \dfrac{5}{2}$

The equation of straight line is given by,

y = mx + c, where m is the slope and c is the y-intercept.

Comparing y = mx + c with y = $\dfrac{-x}{2} + \dfrac{5}{2}$, we get:

⇒ m₁ = $-\dfrac{1}{2}$

Given,

⇒ 2x + 4y + 9 = 0

Converting 2x + 4y + 9 = 0 in the form y = mx + c we get,

⇒ 4y = -2x - 9

⇒ y = $\dfrac{-2x}{4} - \dfrac{9}{4}$

⇒ y = $\dfrac{-1x}{2} - \dfrac{9}{4}$

Comparing y = mx + c with y = $\dfrac{-1x}{2} - \dfrac{9}{4}$, we get:

⇒ m₂ = $-\dfrac{1}{2}$

Since the gradient of the first line is equal to the gradient of the second line.

The lines are parallel to each other.

Hence, proved that lines are parallel.