Question

The equations for two lines in the coordinate plane are 2dx - y = -4 and 4x - y = -6, where d represents an unknown value. What value(s) of d would make these lines perpendicular?

Answer 1

The value of d that makes the lines 2dx - y = -4 and 4x - y = -6 perpendicular is -1/8, as this value makes the product of their slopes equal to -1.

Step-by-step explanation:

To determine which value(s) of d would make the two lines perpendicular, we have to evaluate the slopes of the two lines and set their product to be -1, since perpendicular lines have slopes that are negative reciprocals of each other. Rearranging the first equation, 2dx - y = -4, we get y = 2dx + 4, which has a slope of 2d. For the second equation, 4x - y = -6, rearranging gives y = 4x + 6, with a slope of 4.

The product of the slopes of two perpendicular lines should be -1, so:

(2d) * 4 = -1

Solving for d: d = -1/8.

Thus, when the value of d is -1/8, the two lines are perpendicular.

Answer 2

First, let's put both lines in Slope-intercept form (y = mx + b) to make it a bit easier:
Line A:
2dx - y = -4
-y = -4 - 2dx
y = 2dx + 4

Line B:
4x - y = -6
-y = -6 - 4x
y = 4x + 6

We want a value for d so that line A is perpendicular to line B, which has a slope of 4. That means that the slope of line A must equal -1/4.
2d = -1/4
d = -1/8