Which of the following pairs of linear equations are consistent/inconsistent ? If consistent, obtain the solution graphically :

x + y = 5, 2x + 2y = 10

Given,

Equations :

⇒ x + y = 5, 2x + 2y = 10

⇒ x + y - 5 = 0, 2x + 2y - 10 = 0

Comparing equations x + y - 5 = 0 and 2x + 2y - 10 = 0 with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 respectively, we get :

a₁ = 1, b₁ = 1, c₁ = -5, a₂ = 2, b₂ = 2 and c₂ = -10.

$\dfrac{a$1}{a2} = \dfrac{1}{2}

$\dfrac{b$1}{b2} = \dfrac{1}{2}

$\dfrac{c$1}{c2} = \dfrac{-5}{-10} = \dfrac{1}{2}

Since, $\dfrac{a$1}{a2} = \dfrac{b1}{b2} = \dfrac{c1}{c2}.

Hence, pair of lines are consistent.

1st equation : x + y - 5 = 0

⇒ y = 5 - x …….(1)

2nd equation : 2x + 2y - 10 = 0

⇒ 2y = 10 - 2x

⇒ y = $\dfrac{10 - 2x}{2}$ …….(2)

Table for equation (1) :

Table for equation (2) :

Steps of construction :

1. Plot the points (1, 4), (2, 3) and (3, 2) and join them to form equation (1).

2. Plot the points (1, 4), (2, 3) and (3, 2) and join them to form equation (2).

From graph,

Both the lines are coincident.

Hence, all the points on the co-incident line are solution i.e. infinitely many solutions.