Find the joint equation of the lines passing through the origin and perpendicular to the lines `x + 2y = 19` and `3x + y = 18.`

Explanation

Let `L_(1)` and `L_(2)` be the lines passing through the origin and perpendicular to the lines `x + 2y = 19 and 3x + y = 18` respectively.
Slope of the lines `x + 2y = 19 and 3x + y = 18` are `- (1)/(2) and -(3)/(1) = -3` respectively.
`therefore` slope of the lines `L_(1) and L_(2)` are 2 and `(1)/(3)` respectively.
Since the lines `L_(1) and L_(2)` pass through the origin, their equations are
`y = 2x and y = (1)/(3)x`
i.e., `2x - y = 0 and x - 3y = 0`
`therefore` their combined equation is
`(2x-y)(x-3y) =0`
` therefore 2x^(2) - 6xy - xy + 3y^(2) = 0 therefore 2x^(2) - 7xy + 3y^(2) = 0` .