Find the equation of a line parallel to the line 2x + y - 7 = 0 and passing through the point of intersection of the lines x + y - 4 = 0 and 2x - y = 8.

Simultaneously solving equations :

⇒ x + y - 4 = 0 …….(1)

⇒ 2x - y = 8 ……..(2)

Solving equation (1), we get :

⇒ x = 4 - y ………..(3)

Substituting value of x from (3) in (2), we get :

⇒ 2(4 - y) - y = 8

⇒ 8 - 2y - y = 8

⇒ 8 - 3y = 8

⇒ 3y = 0

⇒ y = 0.

Substituting value of y in (3), we get :

⇒ x = 4 - 0 = 4.

Point of intersection = (4, 0).

Given,

Equation :

⇒ 2x + y - 7 = 0

⇒ y = -2x + 7

Comparing above equation with y = mx + c, we get :

⇒ m = -2.

We know that,

Slope of parallel lines are equal.

∴ Slope of line parallel to 2x + y - 7 is -2.

By point-slope formula,

Equation of line :

⇒ y - y₁ = m(x - x₁)

Substituting value we get :

Equation of line parallel to line 2x + y - 7 = 0 and passing through the point of intersection of the lines x + y - 4 = 0 and 2x - y = 8 is :

⇒ y - 0 = -2(x - 4)

⇒ y = -2x + 8

⇒ 2x + y = 8.

Hence, the equation of required line is 2x + y = 8.