Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

The equation of a line is given. Find the slope of a line that is a. parallel to the line with the given equation; and b. perpendicular to the line with the given equation. $$y=3 x$$

Step-by-Step Solution

Answer

The slope of a line parallel to \(y = 3x\) is 3 and the slope of a line perpendicular to \(y = 3x\) is \(-1/3\).

Step 1: Find the slope of a line parallel to the given line

Slope of a line parallel to the line \(y = 3x\) will be the same as the slope of the given line. From the given equation \(y = 3x\), slope is 3.

Step 2: Find the slope of a line perpendicular to the given line

Perpendicular lines have slopes that are negative reciprocals of each other. Hence, the slope of the line perpendicular to the line \(y = 3x\) will be the negative reciprocal of 3, which calculates to \(-1/3\).

Key Concepts

Parallel Lines

Parallel lines are fascinating in geometry because they run alongside each other and never meet, no matter how far they are extended into infinity. The most important characteristic of parallel lines is that they have the same slope. This means if one line is described by the equation \( y = mx + b \), then any line parallel to it will also have the slope \( m \), but may have a different y-intercept, described by \( b \).
Let's look at the equation \( y = 3x \). This line has a slope of 3. Consequently, every line that is parallel to this one must also have a slope of 3. The y-intercept of the line, however, can vary, allowing parallel lines to occupy different positions on the graph without ever intersecting. Thus, while the slope keeps the lines consistently parallel, the y-intercept gives each line its unique place.
Understanding this concept is essential not only in solving mathematical problems but also in visualizing how parallel lines behave in various fields, from road engineering to computer graphics.

Perpendicular Lines

Perpendicular lines introduce a new dynamic by intersecting at right angles (90 degrees). When two lines are perpendicular, their slopes have a special relationship. Specifically, their slopes are negative reciprocals of each other. This means if one line has a slope of \( m \), the other will have a slope of \( -\frac{1}{m} \).
To illustrate, consider the equation \( y = 3x \). Here, the slope is 3. Therefore, any line that is perpendicular to this one must have a slope that is the negative reciprocal of 3, which computes to \(-\frac{1}{3}\). This relationship ensures that at the point of intersection, the two lines form perfect right angles, which can be crucial in fields such as architecture and engineering.
Grasping the concept of perpendicularity helps not only in solving equations but also in applications where right angles are a necessity. This principle is a fundamental pillar for constructing geometric shapes and houses, making it a vital tool in various practical contexts.

Equation of a Line

The equation of a line is a mathematical statement that describes a line's trajectory across a graph. Typically expressed in the slope-intercept form \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. It's essential to understand what these terms mean.

• Slope (\( m \)): Indicates how steep the line is and the direction it tilts. A positive slope means the line ascends from left to right, while a negative slope implies it descends.
• Y-Intercept (\( b \)): The point where the line crosses the y-axis. It represents the value of \( y \) when \( x \) is zero.

With these components, you can predict how a line behaves for any given value of \( x \). For example, in the line equation \( y = 3x \), the slope is 3, meaning it rises fairly steeply as it moves to the right, and the y-intercept is 0, indicating it passes exactly through the origin.
Mastering the equation of a line is crucial for creating graphs and solving systems of equations, both of which are fundamental skills in algebra and calculus.