Conditions for pair of linear equations are consisten
`(a_(1))/(a_(2)) != (b_(1))/(b_(2))" " ` [ unique solution]
and `" " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2)) " " ` [ infinitely many solutions ]
(i) No, the given pair of linear equations
`-3x-4y = 12 ` and ` 3x + 4y = 12`
Here, `" " a_(1) = -3, b_(1) = -4, c_(1) = -12`,
`a_(2) = 3, b_(2) = 4, c_(2) = -12`
Now, `(a_(1))/(a_(2)) = -(3)/(3) = -1, (b_(1))/(b_(2))= - (4)/(4) = -1, (c_(1))/(c_(2))= (-12)/(-12) = 1`
`:' " " (a_(1))/(a_(2)) = (b_(1))/(b_(2))!= (c_(1))/(c_(2))`
Hence, the pair of linear equations has no solution, i. e., inconsisten.
(ii) Yes, the given pair of linear equations
`(3)/(5)x - y = (1)/(2) ` and ` (1)/(5)x - 3y = (1)/(6)`
Here, `" " a_(1) = (3)/(5), b_(1) = - 1, c_(1) = -(1)/(2)`
and `" " a_(2) = (1)/(5), b_(2) = -3, c_(2) = -(1)/(6)`
Now, `" " (a_(1))/(a_(2)) = (3)/(1), (b_(1))/(b_(2)) = (-1)/(-3) = (1)/(3) , (c_(1))/(c_(2)) = (3)/(1) " " [ :' (a_(1))/(a_(2)) != (b_(1))/(b_(2))]`
Hence, the given pair of linear equations has unique solution, i.e., consisten.
(iii) Yes, the given pair of linear equations
` 2ax + by - a = 0`
and `" " 4ax + 2by - 2a = 0, a, b != 0`
Here, `" " a_(1) = 2a, b_(1) = b, c_(1) = -a`,
`" " a_(2) = 4a, b_(2) = 2b, c_(2) = -2a`
Now, `" " (a_(1))/(a_(2)) = (2a)/(4a) = (1)/(2), (b_(1))/(b_(2)) = (b)/ (2b) = (1)/(2), (c_(1))/(c_(2))= (-a)/(-2a) = (1)/(2)`
`:' " " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))= (1)/(2)`
Hence, the given pair of linear equations has infinitely many solutions, i.e., consistent or dependent.
(iv) no, the given pair of linear equations
x + 3y = 11 and 2x + 6y = 11
Here, `" " a_(1) = 1, b_(1) = 3, c_(1) = -11 " " ...(i) `
`" " a_(2) = 2, b_(2) = 6, c_(2) = -11`
Now, `" " (a_(1))/(a_(2)) = (1)/(2), (b_(1))/(b_(2)) = (3)/(6), (1)/(2), (c_(1))/(c_(2)) = (-11)/(-11) = 1`
`:. " " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) != (c_(1))/(c_(2))`
Hence, the pair of linear equation have no solution i.e., inconsistent.
