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) 3x – 5y = 20 and 6x – 10y = 40
(ii) x – 3y – 7 = 0 and 3x – 3y – 15 = 0
(i) Given, 3x – 5y = 20 and 6x – 10y = 40
(a1/a2) = 3/6 = 1/2
(b1/b2) = -5/-10 = 1/2
(c1/c2) = 20/40 = 1/2
a1/a2 = b1/b2 = c1/c2
Since the given sets of lines are overlapping each other there will be infinite number of solutions for this pair of equation.
(ii) Given, x – 3y – 7 = 0 and 3x – 3y – 15 = 0
(a1/a2) = 1/3
(b1/b2) = -3/-3 = 1
(c1/c2) = -7/-15
a1/a2 ≠ b1/b2
Since this pair of lines are intersecting each other at a unique point, there will be a unique solution.
By cross multiplication,
x/(45-21) = y/(-21+15) = 1/(-3+9)
x/24 = y/ -6 = 1/6
x/24 = 1/6 and y/-6 = 1/6
∴ x = 4 and y = 1.