Do the following equations represent a pair of coincident lines ? Justify your answer.
(i) `3x + (1)/(7)y = 3` and `7x + 3y = 7`
(ii) `-2x - 3y = 1 ` and ` 6y + 4x = -2`
(iii) `(x)/(2) + y + (2)/(5) = 0` and `4x + 8y + (5)/(16) = 0`

Explanation

Condition for coincident lines,
`(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
(i) No, given pair of linear equations
`3x + (y)/(7) - 3 = 0`
and `" " 7x + 3y - 7 = 0`,
where, `" " a_(1) = 3, b_(1) = (1)/(7), c_(1) = -3 `,
`a_(2) = 7, b_(2) = 3, c_(2) = -7`
Now, `" " (a_(1))/(a_(2)) = (3)/(7), (b_(1))/(b_(2)) = (1)/(21), (c_(1))/(c_(2)) = (3)/(7) " " [ :' (a_(1))/(a_(2)) != (b_(1))/(b_(2))]`
Hence, the given pair of linear equations has unique solution.
(ii) Yes, given pair of linear equations
`-2x - 3y - 1 = 0 ` and ` 6y + 4x + 2 = 0 `
where, `" " a_(1) = -2, b_(1) = -3, c_(1) = -1`,
`" " a_(2) = 4, b_(2) = 6, c_(2) = 2`
Now, `" " (a_(1))/(a_(2))=-(2)/(4) = - (1)/(2)`
`" " (b_(1))/(b_(2)) = - (3)/(6) = - (1)/(2), (c_(1))/(c_(2)) = - (1)/(2)`
` :' " " (a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2)) = -(1)/(2)`
Hence, the given pair of linear equations is coincident
(iii) No, the given pair of linear equations are
`(x)/(2) + y + (2)/(5) = 0 ` and ` 4x + 8y + (5)/(16) = 0`
Here, `" " a_(1) = (1)/(2), b_(1) = 1, c_(1) = (2)/(5)`
`" " a_(2) = 4, b_(2) = 8, c_(2) = (5)/(16)`
Now, `" " (a_(1))/(a_(2)) = (1)/(8), (b_(1))/(b_(2)) = (1)/(8), (c_(1))/(c_(2)) = (32)/(25)`
`:' " " (a_(1))/(a_(2))=(b_(1))/(b_(2)) != (c_(1))/(c_(2))`
Hence, the given pair of linear equations has no solution.