# For what value of a are the following lines parallel. Can the value be negative. a^2x - 3y + 8 = 0, 3x + ay + 9 = 0

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The value of a has to be determined for which the lines a^2x - 3y + 8 = 0 and 3x + ay + 9 = 0 are parallel.

Parallel lines have equal slope. Express the two lines in the slope intercept form, y = mx + b

a^2x - 3y + 8 = 0

=> 3y = a^2x + 8

=> `y = (a^2/3)x + 8/3`

The slope of this line is `a^2/3`

3x + ay + 9 = 0

=> -ay = 3x + 9

=> `y = (-3/a)x - 9/a`

The slope of this line isĀ `-3/a`

Equating the slope of the lines `a^2/3 = -3/a`

=> `a^3 = -9`

`a = -root(3) 9 ~~ -2.08`

**The value of a for which the lines are parallel is `-root(3) 9` .**

This value can be negative. The slope of a line can be of either sign, a negative value indicates a decrease in the value of y for an increase in the value of x.