Question

Determine which of the lines, if any, are parallel or perpendicular. Explain.

Line a: 4y - 12x = 20
Line b: 3y= -x + 2
Line c: y= 3x - 1

Answer 1

Line
`a` and line
`c` are parallel to each other.

Line
`b` is perpendicular to line
`a`.

Line
`b` is perpendicular to line
`c`.

Explanation:

Rewrite the equation of each line in the slope-intercept form
`y = m\, x + b` to find the slope
`m` of that line.

In the slope-intercept form,
`y` need to be on the left-hand side of the equation while
`x` need to be on the right-hand side. The coefficient of
`y` must be
`1`. The slope of the line would then be equal to the coefficient of
`x`.

For example, the equation of line
`a` could be rewritten in slope-intercept form as
`y = 3\, x + 5`. The coefficient of
`x` is
`3`, so the slope of this line would be
`m_(a) = 3\!`.

Similarly, rewrite to obtain the the slope-intercept equation of line
`b`:
`y = (-1/3)\, x + (2/3)`. The slope of this line would be
`m_(b) = (-1/3)`.

The equation of line
`c` is already in slope-intercept form. The slope of that line would be
`m_(c) = 3`.

Let
`m_(1)` and
`m_(2)` denote the slopes of two lines in a Cartesian plane.

• These two lines are parallel with if and only if their slopes are the same:
  `m_(1) = m_(2)`.
• These two lines are perpendicular to each other if the product of their slope is
  `(-1)`:
  `m_(1)\, m_(2) = (-1)`.

In this question, the slope of line
`a` and line
`c` are the same:
`m_(a) = m_(c)`. Thus, line
`a\!` and line
`c\!` would be parallel.

The product of the slopes of line
`a` and line
`b` is
`m_(a)\, m_(b) = 3 * (-1/3) = (-1)`. Thus, line
`a\!` and line
`b` are perpendicular to each other.

Similarly, line
`b` and line
`c` are perpendicular to each other because the product of their slopes is
`(-1)`.