Condition for no solution ` (a_(1))/(a_(2)) = (b_(1))/(b_(2)) != (c_(1))/(c_(2))`
(i) Yes, given pair of equations,
`2x + 4y = 3` and ` 12y + 6x = 6`
Here, `" " a_(1) = 2, b_(1) = 4, c_(1) = -3`,
`a_(2) = 6, b_(2) = 12, c_(2) = -6`
`:. " " (a_(1))/(a_(2)) = (2)/(6) = (1)/(3) , (b_(1))/(b_(2)) = (4)/(12) = (1)/(3)`
`" " (c_(1))/(c_(2)) = (-3)/(-6) = (1)/(2) `
`:. " " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) != (c_(1))/(c_(2))`
Hence, the given pair of linear equations has no solution.
(ii) No, given pair of equations,
x = 2y and y = 2x
or `" " ` x - 2y = 0 and 2x - y = 0
Here, `" " a_(1) = 1, b_(1) = -2, c_(1) = 0`,
`" " a_(2) = 2, b_(2) = -1, c_(2) = 0`
`:. " " (a_(1))/(a_(2)) = (1)/(2) ` and `(b_(1))/(b_(2)) = (2)/(1) `
`:' " " (a_(1))/(a_(2)) != (b_(1))/(b_(2))`
Hence, the given pair of linear equations has unique solution.
(iii) No, given pair of equations, ` " " 3x y - 3 = 0 ` and ` 2x + (2)/(3) y - 2 = 0`
Here, `" " a_(1) = 3, b_(1) = 1, c_(1) = -3`,
`" " a_(2) = 2, b_(2) = (2)/(3) , c_(2) = -2`
`:. " " (a_(1))/(a_(2)) = (3)/(2), (b_(1))/(b_(2)) = (1)/ (2//3) = (3)/(2) `
`" " (c_(1))/(c_(2)) = (-3)/(-2) = (3)/(2)`
`:' " " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2)) = (3)/(2)`
Hence, the given pair of linear equations is coincident and having infinitely many solutions.
