Indicate whether the lines are parallel, perpendicular, or neither. justify your answer.

a. y=4x-1 and 12x=3y+7
b. x-7y=10 and 2x+14y=21
c. 5x+6y=18 and 18x-15y=36
d. x=-1 and y=-1

Note that when two lines have slopes of m₁ and m₂, then
(i) If m₁ = m₂, then the lines are parallel.
(ii) If m₁*m₂ = -1, then the lines are perpendicular.

Let us evaluate the given lines.
a. y = 4x - 1 and 12x = 3y + 7
    y = 4x - 1  => m₁ = 4
    Write the second equation in standard form.
    3y = 12x - 7
      y = 4x - 7/3 => m₂ = 4.
    m₁ = m₂.
    PARALLEL

b. x - 7y = 10 and 2x + 14y = 21
   In standard form,
   7y = x - 10 or y = (1/7)x - 10/7      => m₁ = 1/7
   14y = - 2x + 21  or y = -1/7 + 3/2  => m₂ = -1/7
   m₁ ≠ m₂ and m₁*m₂ ≠ -1.
    NEITHER

c. 5x + 6y = 18 and 18x - 15y = 36
    In standard form,
    6y = -5x + 18 or  y = -(5/6)x + 3     => m₁ = -5/6
    15y = 18x - 36 or y = (6/5)x - 12/5  => m₂ = 6/5
    m₁*m₂ = -1
    PERPENDICULAR

d. x =1 and y = 1
    x = 1 => m₁ is undefined
    y = 1  => m₂ = 0
    m₁ ≠ m₂ and m₁*m₂ ≠ -1
    NEITHER

The correct answer is: b

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