Problem 36 - College Algebra

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. \(y=2 x+7\) \(y=-\frac{1}{2} x-4\)

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Identify the slopes

The given equations are in the slope-intercept form, which is given by \(y = mx + c\), where \(m\) is the slope. For the first equation \(y = 2x + 7\), the slope is \(2\). For the second equation \(y = -\frac{1}{2}x - 4\), the slope is \(-\frac{1}{2}\).

Step 2: Determine the criteria for parallel lines

For two lines to be parallel, their slopes must be equal. From Step 1, the slopes are \(2\) and \(-\frac{1}{2}\). Since these are not equal, the lines are not parallel.

Step 3: Determine the criteria for perpendicular lines

Two lines are perpendicular if the product of their slopes is \(-1\). Multiply the slopes obtained in Step 1: \(2 \times -\frac{1}{2} = -1\). Since the product is \(-1\), the lines are perpendicular.

Key Concepts

Slope-Intercept Form

To graph a linear equation easily and quickly, we can use the slope-intercept form. This form of a linear equation is expressed as \(y = mx + c\). In this formula, \(m\) represents the slope of the line, and \(c\) denotes the y-intercept, which is where the line crosses the y-axis.
This form is especially useful when graphing because you can directly see how steep the line is and where it starts on the y-axis. Knowing these two components, the slope and y-intercept, you can plot the line accurately.
For example, with the equation \(y = 2x + 7\), the slope \(m\) is 2, and the y-intercept \(c\) is 7. This means the line rises steeply as it crosses the y-axis at \((0, 7)\). Similarly, for \(y = -\frac{1}{2}x - 4\), the line falls as it crosses the y-axis at \((0, -4)\).

Parallel Lines

Parallel lines in a graph have a special characteristic: they never meet or intersect, no matter how far they are extended on a plane.
For two lines to be parallel, they must have the same slope. If the slopes of two lines are identical, they will always have the same steepness and direction, hence never crossing each other.
In the equations from our problem, \(y = 2x + 7\) and \(y = -\frac{1}{2}x - 4\), the slopes are 2 and \(-\frac{1}{2}\), respectively. Since these values are not equal, these lines are not parallel.
To check if two given lines are parallel, always compare their slopes. If they are the same, then the lines are parallel.

Perpendicular Lines

In contrast to parallel lines, perpendicular lines intersect at a right angle (90 degrees). The key property of perpendicular lines is the relationship between their slopes.
For two lines to be perpendicular, the product of their slopes must be \(-1\). This means when you multiply the slopes of the two lines, the result should be \(-1\).
Given \(y = 2x + 7\) and \(y = -\frac{1}{2}x - 4\), we find the slopes are 2 and \(-\frac{1}{2}\). If we multiply these slopes together: \(2 \times -\frac{1}{2} = -1\). Thus, the lines are perpendicular.
Whenever you are trying to determine if two lines are perpendicular, remember to multiply their slopes. If the result is \(-1\), the lines will definitely intersect at a 90-degree angle.

Slope of a Line

The slope of a line represents how steep the line is on a graph. It is a measure of the line's tilt or angle.
In the slope-intercept form \(y = mx + c\), the slope is denoted by \(m\). It tells you how much the y-coordinate (vertical) changes as the x-coordinate (horizontal) increases. A positive slope means the line rises as it goes from left to right, while a negative slope indicates it falls.
For our equations, \(y = 2x + 7\) has a slope of 2. This slope indicates the line rises steeply, moving two units up for every one unit across. For \(y = -\frac{1}{2}x - 4\), the slope \(-\frac{1}{2}\) shows the line falls, dropping a half unit for every one unit it moves horizontally.
Understanding the slope's value is crucial when graphing a line, determining parallelism, or evaluating the perpendicularity between lines.