Question

Given that the two of the three lines 3x+4y=5,5x+By=13, and 7x=2425 are perpendicular, compute B. Express your answer as a fraction in simplest form.

Answer 1

B = -¹⁵/₄

Explanation:

The slope of 7x = 2425 is 0, and the other two lines have non-zero slopes, so they must be the ones that are perpendicular.

(1) 3x + 4y = 5

(2) 5x + By = 13

1. Calculate the slope of Line (1)

3x + 4y = 5

Subtract 3x from each side: 4y = -3x + 5

Divide each side by 4: y = -¾x + ⁵/₄

The slope m₁ = -¾.

2. Calculate the slope of the perpendicular line

m₂ = -1/m₁

m₂ = ⁴/₃

3. Calculate the slope of Line (2)

5x + By = 13

Subtract 5x from each side: By = -5x + 13

Divide each side by B: y = (-5/B)x + 13/B

The slope m = -5/B.

4. Calculate the value of B

If Line (2) is the perpendicular line, then

⁴/₃ = -5/B

Multiply each side by B: (⁴/₃)B = -5

Multiply each side by 3: 4B = -15

Divide each side by 4: B = -¹⁵/₄

So, the value of B = -¹⁵/₄, and the equation of Line (2) is

5x - (¹⁵/₄)y = 13

The Figure below shows the graph of Line (1) in purple, and the graph of its perpendicular Line (2) in black.