The line 2y+7x=5 and 4x+dy+3=0 are perpendicular to each other. Find the value of the constant 'd'. (5marks)

7X + 2Y = 5 and 4X + dY = 0

m1 = -A/B = -7/2.
m2 = 2/7 = neg. reciprocal of m1.
m2 = -A/B.
2/7 = --4/d
2d = -28
d = -14.

To determine the value of the constant 'd' in the equation 4x + dy + 3 = 0, we will make use of the fact that two lines are perpendicular if the product of their slopes is equal to -1.

First, we need to rearrange both equations in the standard form of a linear equation (y = mx + b), where 'm' represents the slope of the line.

For the equation 2y + 7x = 5:
1. Subtract 7x from both sides: 2y = -7x + 5
2. Divide by 2: y = (-7/2)x + 5/2

The slope of this line is -7/2.

For the equation 4x + dy + 3 = 0, we need to express it in the form y = mx + b by isolating the term with 'y' on one side:
1. Subtract 4x and 3 from both sides: dy = -4x - 3
2. Divide by 'd': y = (-4/d)x - 3/d

The slope of this line is -4/d.

Since the two lines are perpendicular, we can use the fact that the product of their slopes is equal to -1:

(-7/2) * (-4/d) = -1

To solve this equation for 'd', we can multiply both sides of the equation by 2d:

(2d)*(-7/2)*(-4/d) = (-1)*(2d)
-28 = -2d

Divide both sides by -2:
d = 14

Therefore, the value of the constant 'd' is 14.