Problem 42 - 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. $$3 x-4 y=-7$$

Step-by-Step Solution

Answer

The slope of a line parallel to the given equation is \(\frac{3}{4}\), and the slope of a line perpendicular to the given equation is \(-\frac{4}{3}\).

Step 1: Converting Given Equation to Slope-Intercept Form

Rearrange the given equation 3x - 4y = -7 to y = mx + c form. Doing so gives: \[ y = \frac{3}{4}x + \frac{7}{4} \]. Here, m, which is the slope of the given line, is \(\frac{3}{4}\).

Step 2: Finding Parallel Line's Slope

For a line to be parallel to another, their slopes need to be equal. Therefore, the slope of a line parallel to the given line is also \(\frac{3}{4}\).

Step 3: Finding Perpendicular Line's Slope

For a line to be perpendicular to another, its slope is the negative reciprocal of the other line's slope. Hence, the slope of a line perpendicular to the line given in the question is the negative reciprocal of \(\frac{3}{4}\), which is \(-\frac{4}{3}\).

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never intersect or touch each other. They are always the same distance apart.
One of the key characteristics of parallel lines is that they have the same slope. This means if you have one line with a certain slope, any line parallel to it will share that same slope.

• If the slope of one line is represented as \( m \), then the slope of a line parallel to it will also be \( m \).
• For example, if the slope of a given line is \( \frac{3}{4} \), then a parallel line will also have a slope of \( \frac{3}{4} \).

This concept is particularly useful when you need to determine if two lines are parallel just by comparing their slopes. It helps in solving problems involving equations of lines and drafting parallel designs.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which means they meet to form a \( 90^\circ \) angle.
An important feature in mathematics of perpendicular lines is their slopes. The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
Here's how you can find the slope of a perpendicular line:

• If a line has a slope \( m \), then the slope of the line perpendicular to it will be \( -\frac{1}{m} \).
• For example, if the original line’s slope is \( \frac{3}{4} \), then the perpendicular line’s slope will be \( -\frac{4}{3} \).

Understanding perpendicular slopes is essential when analyzing geometric figures and solving various types of equation problems. Moreover, it plays a crucial role in understanding symmetry and construction in architecture.

Slope-Intercept Form

The slope-intercept form of a line is a way to express the equation of a line, making it easy to identify both the slope and the y-intercept directly from the equation.
The general formula is \( y = mx + c \), where:

• \( m \) denotes the slope of the line.
• \( c \) represents the y-intercept, which is the point where the line crosses the y-axis.

By converting any given line equation into this form, like the equation \( 3x - 4y = -7 \), you can easily find that its slope is \( \frac{3}{4} \).
This form is particularly advantageous because it allows you to quickly understand the direction and steepness of the line, as well as where it intersects the y-axis. Slope-intercept form is a foundational concept in algebra, often serving as a primary tool for graphing lines and understanding linear relationships.