Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.
(i) x – 3y – 3 = 0 and 3x – 9y – 2 = 0
(ii) 2x + y = 5 and 3x + 2y = 8
(i) Given, x – 3y – 3 =0 and 3x – 9y -2 =0
a1/a2 = 1/3, b1/b2 = -3/-9 = 1/3, c1/c2 =-3/-2 = 3/2
(a1/a2) = (b1/b2) ≠ (c1/c2)
Since, the given set of lines are parallel to each other they will not intersect each other and therefore there is no solution for these equations.
(ii) Given, 2x + y = 5 and 3x +2y = 8
a1/a2 = 2/3, b1/b2 = 1/2, c1/c2 = -5/-8
(a1/a2) ≠ (b1/b2)
Since they intersect at a unique point these equations will have a unique solution by cross multiplication method:
x/(b1c2-c1b2) = y/(c1a2 – c2a=) = 1/(a1b2-a2b1)
x/(-8-(-10)) = y/(15+16) = 1/(4-3)
x/2 = y/1 = 1
∴ x = 2 and y =1