Question

State which pairs of lines are:(a) Parallel to each other.(b) Perpendicular to each other.

Answer 1

So first of all we should write the three equations in slope-intercept form. This will make the problem easier to solve. Remember that the slope-interception form of an equation of a line looks like this:

`y=mx+b`

Where m is known as the slope and b the y-intercept. The next step is to rewrite the second and third equation since the first equation is already in slope-intercept form. Its slope is 4 and its y-intercept is -1.

So let's rewrite equation (ii). We can begin with substracting 4 from both sides of the equation:

`\begin{gathered} 8y+4=-2x \\ 8y+4-4=-2x-4 \\ 8y=-2x-4 \end{gathered}`

Then we can divide both sides by 8:

`\begin{gathered} (8y)/(8)=(-2x-4)/(8) \\ y=-(2)/(8)x-(4)/(8) \\ y=-(1)/(4)x-(1)/(2) \end{gathered}`

So its slope is -1/4 and its y-intercept is -1/2.

For equation (iii) we can add 8x at both sides:

`\begin{gathered} 2y-8x=-2 \\ 2y-8x+8x=-2+8x \\ 2y=8x-2 \end{gathered}`

Then we can divide both sides by 2:

`\begin{gathered} (2y)/(2)=(8x-2)/(2) \\ y=(8)/(2)x-(2)/(2) \\ y=4x-1 \end{gathered}`

Then its slope is 4 and its y-intercept is -1. As you can see this equation is equal to equation (i).

In summary, the three equations in slope-intercept form are:

`\begin{gathered} (i)\text{ }y=4x-1 \\ (ii)\text{ }y=-(1)/(4)x-(1)/(2) \\ (iii)\text{ }y=4x-1 \end{gathered}`

It's important to write them in this form because when trying to figure out if two lines are parallel or perpendicular we have to look at their slopes:

- Two lines are parallel to each other if they have the same slope (independently of their y-intercept).

- Two lines are perpendicular to each other when the slope of one of them is the inverse of the other multiplied by -1. What does this mean? If a line has a slope m then a perpendicular line will have a slope:

`-(1)/(m)`

Now that we know how to find if two lines are parallel or perpendicular we can find the answers to question 4.

So for part (a) we must find the pairs of parallel lines. As I stated before we have to look for those lines with the same slope. As you can see, only lines (i) and (iii) have the same slope (4) so the answer to part (a) is: Lines (i) and (iii) are parallel to each other.

For part (b) we have to look for perpendicular lines. (i) and (iii) are parallel so they can't be perpendicular. Their slopes are equal to 4 so any line perpendicular to them must have a slope equal to:

`-(1)/(m)=-(1)/(4)`

Which is the slope of line (ii). Then the answer to part (b) is that lines (i) and (ii) are perpendicular to each other as well as lines (ii) and (iii).