Problem 102 - Algebra for College Students

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. $$9-x=3, \frac{1}{2} x=8$$

Step-by-Step Solution

Answer

The lines are parallel.

Step 1 - Convert the equations into slope-intercept form

First, convert both equations to the slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2 - Convert the first equation

Starting with the first line, the equation is given as \( 9 - x = 3 \).Subtract 9 from both sides: \( -x = 3 - 9 \).This simplifies to \( -x = -6 \).Multiply by -1 to get \( x = 6 \), which represents the line in the form \( x = 6 \).This is a vertical line, so its slope is undefined.

Step 3 - Convert the second equation

Now, for the second equation, \( \frac{1}{2} x = 8 \).To find \( y \), rewrite the equation in terms of \( y = mx + b \).First, solve for \( x \): \( x = 16 \).This can be considered as \( x = 16 \), which again is a vertical line with an undefined slope.

Step 4 - Determine the relationship between the slopes

For lines to be parallel, their slopes must be equal. For lines to be perpendicular, the product of their slopes must be -1. Since both lines have undefined slopes and thus are vertical lines, they cannot be perpendicular.However, two vertical lines are parallel to each other.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a very handy method for understanding lines on a graph. It is given by the formula: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The slope tells you how steep the line is, while the y-intercept is where the line crosses the y-axis.
For example, if you have the equation \( y = 2x + 3 \):

• The slope \( m \) is 2, meaning for every step you go to the right on the x-axis, you go up 2 steps on the y-axis.
• The y-intercept is 3, meaning the line crosses the y-axis at \( y = 3 \).

In the exercise, converting equations to this form helps us easily identify their slopes and other properties.

Vertical Lines

A vertical line is a special type of line where all points have the same x-coordinate. The equation of a vertical line is generally represented as \( x = a \) where \( a \) is a constant. In our exercise:

• The equation \( x = 6 \) means it is a vertical line passing through \( x = 6 \).
• The equation \( x = 16 \) is another vertical line passing through \( x = 16 \).

Vertical lines are interesting because they don't have a well-defined slope like other lines. Instead, their slope is considered undefined.

Undefined Slope

The concept of slope usually tells us how steep a line is. For a line that runs diagonally, we can calculate the slope using the formula: \[ m = \frac{ \text{rise} }{ \text{run} } \] However, when we talk about vertical lines, things change. The 'run' (or the change in x) for a vertical line is 0 because the x-coordinate never changes. This leads to a situation where we are dividing by zero, which is not defined in mathematics. Hence, we say the slope is 'undefined'.
In summary, vertical lines like \( x = 6 \) and \( x = 16 \) from our exercise have undefined slopes, and they will appear as straight up-and-down lines when graphed.
This also explains why vertical lines are always parallel to each other; they never cross or meet at any point. They share the same undefined slope and run in the same direction.