Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 16
(i) Given, x + y = 5 and 2x + 2y = 10
(a1/a2) = 1/2
(b1/b2) = 1/2
(c1/c2) = 1/2
Since (a1/a2) = (b1/b2) = (c1/c2)
∴ The equations are coincident and they have infinite number of possible solutions.
So, the equations are consistent.
For, x + y = 5 or x = 5 – y
x → 4, 3, 2
y → 1, 2, 3
For 2x + 2y = 10 or x = (10-2y)/2
x → 4, 3, 2
y → 1, 2, 3
So, the equations are represented in graphs as follows:
(ii) Given, x – y = 8 and 3x – 3y = 16
(a1/a2) = 1/3
(b1/b2) = -1/-3 = 1/3
(c1/c2) = 8/16 = 1/2
Since, (a1/a2) = (b1/b2) ≠ (c1/c2)
The equations are parallel to each other and have no solutions. Hence, the pair of linear equations is inconsistent.