Problem 48 - Contemporary Precalculus

Question

Determine whether the lines whose equations are given are parallel, perpendicular, or neither. \(3 x+y-3=0 \quad\) and \(\quad 6 x+2 y+17=0\)

Step-by-Step Solution

Answer

Lines: 1. 3x + y - 3 = 0 2. 6x + 2y + 17 = 0 Answer: The given lines are neither parallel nor perpendicular.

Step 1: Write the equations in slope-intercept form

To easily compare the slopes, rewrite both given equations in the slope-intercept form, which is \(y = mx + b\) where m is the slope and b is the y-intercept. Equation 1: \(3x + y - 3 = 0\) To solve for y, perform these steps: 1. Subtract \(3x\) from both sides: \(y - 3x = 3\) 2. Add 3x to both sides: \(y = 3x + 3\) Equation 2: \(6x + 2y + 17 = 0\) To solve for y, perform these steps: 1. Subtract \(6x\) from both sides: \(2y = -6x - 17\) 2. Divide both sides by 2: \(y = -3x - \frac{17}{2}\)

Step 2: Compare the slopes

Now, compare the slopes of the two equations: Equation 1: Slope = 3 Equation 2: Slope = -3 Since the slopes are neither the same nor negative reciprocals, the lines are neither parallel nor perpendicular.

Key Concepts

Parallel and Perpendicular Lines

Understanding the relationship between two lines includes identifying whether they are parallel, perpendicular, or neither.

Parallel lines are two or more lines that never intersect because they have the same slope but different y-intercepts. For example, the lines represented by the equations \(y = 2x + 3\) and \(y = 2x - 4\) are parallel since both have a slope of 2.

Perpendicular lines intersect at a 90-degree angle, and their slopes are negative reciprocals of each other. That means if one line has a slope of \(m\), the other will have a slope of \(-\frac{1}{m}\). An example of this is the lines given by \(y = \frac{1}{2}x + 1\) and \(y = -2x + 3\), where the slopes are 1/2 and -2, respectively.

When analyzing the given exercise equations, we find that the slopes are 3 and -3, which are not equal nor negative reciprocals of each other. Hence, the lines are neither parallel nor perpendicular, offering a clear example of how these concepts apply in practice.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is calculated as the 'rise over run', indicating how many units the line goes up (or down) for each unit it moves to the right.

In mathematical terms, the slope is often represented as 'm' in the slope-intercept equation \(y = mx + b\), where \(m\) denotes the slope. To determine the slope from an equation, we need to solve it for \(y\) to get the slope-intercept form. From the exercise, the first equation was transformed into \(y = 3x + 3\), revealing a slope of 3. The second equation became \(y = -3x - \frac{17}{2}\), with a slope of -3.

We expect parallel lines to share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. However, in this case, with slopes 3 and -3 for the two given lines, these slopes indicate neither parallel nor perpendicular orientations, illustrating the crucial role of slope in determining the relationship between lines.

Slope-Intercept Equation

The slope-intercept equation is a fundamental representation of a straight line in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.

To visualize this, consider the graph of a line with equation \(y = mx + b\). The slope, \(m\), dictates how the line tilts upwards or downwards as it moves from left to right, while the y-intercept, \(b\), pinpoints the line's starting position on the y-axis. The exercise provided demonstrates the process of converting standard form equations into slope-intercept form to easily extract these parameters.

After performing the conversion, we get the equations in the desired form: \(y = 3x + 3\) and \(y = -3x - \frac{17}{2}\). Through this form, we immediately recognize the slopes and y-intercepts, enabling us to analyze the line's characteristics and relationships quickly.