What is the equation of a line perpendicular to 6(2x+12)=4y that passes through the origin?

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First, let's open the parentheses in the given equation: `1 2 x + 7 2 = 4 y . ` Then reduce by the factor `4 : ` `3 x + 1 8 = y . ` Write it in the form `A x + B y = C :`

`3 x - y = - 1 8 .`

The equation of a perpendicular line has the form `x + 3 y = C ,` because their normal (orthogonal) vectors are `( 3 , - 1 ) ` and `( 1 , 3 ) ,` which are perpendicular (one can use dot product to verify this).

Finally, it is given that the orthogonal line passes through the origin: that is, the point with `x = y = 0 ` satisfies the equation `x + 3 y = C .` This immediately shows that `C = 0 ` and the resulting equation is `x + 3 y = 0 . ` It may be also written as `y = - 1 / 3 x `, if you wish.

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