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Chapter 3: Problem 3

Determine the slope of parallel lines and perpendicular lines. $$ y=4 x+4 $$

Short Answer

Expert verified
Parallel: 4; Perpendicular: -1/4

Step by step solution

01

Identify Given Line's Slope

First, identify the slope of the given line, which is posed in slope-intercept form (\(y = mx + c\)). Here, the coefficient of \(x\) is the slope, \(m = 4\).
02

Determine Parallel Line Slope

For a line to be parallel to another, both lines must have the exact same slope. Therefore, any line parallel to \(y = 4x + 4\) must also have a slope of \(4\).
03

Determine Perpendicular Line Slope

For a line to be perpendicular to another, the slopes must be negative reciprocals of each other. If the given line's slope is \(4\) (which can be expressed as \(\frac{4}{1}\)), the negative reciprocal is \(-\frac{1}{4}\). Hence, any line perpendicular to \(y = 4x + 4\) will have a slope of \(-\frac{1}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
When two lines in a plane do not intersect, they are considered parallel. This happens if and only if they have the same slope. In mathematical terms, if you have a line in the slope-intercept form given by \( y = mx + c \), any other line that is parallel to it will also have the slope \( m \). However, the \( c \) (or y-intercept) can be different, meaning the lines can be at different heights on the graph, but they will never cross each other.
For example, with the line \( y = 4x + 4 \), any line that is parallel must have a slope of \( 4 \). So you may encounter equations like \( y = 4x + 1 \) or \( y = 4x + 10 \), which are parallel as they have identical slopes.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). To achieve this, the slopes of perpendicular lines are negative reciprocals of each other. This means if the slope of one line is \( m \), the slope of a line perpendicular to it would be \( -1/m \).
For instance, consider a line with a slope of \( 4 \) such as \( y = 4x + 4 \). The negative reciprocal of \( 4 \) is \(-\frac{1}{4}\). Therefore, any line with a slope of \(-\frac{1}{4}\) will be perpendicular to the line \( y = 4x + 4 \). Such perpendicular lines visually form a perfect 'T' shape when they intersect.
Slope-Intercept Form
The slope-intercept form is a simple yet powerful way to express linear equations. It is written as \( y = mx + c \), where:
  • \( m \) represents the slope of the line. It shows the steepness and direction.
  • \( c \) is the y-intercept. This is the point where the line crosses the y-axis.
Using this form, you can easily identify the key characteristics of a line. For example, with the equation \( y = 4x + 4 \), the slope \( m \) is \( 4 \), meaning the line rises 4 units vertically for every unit it moves horizontally. This indicates a relatively steep slope. The y-intercept \( c \) is \( 4 \), indicating the line crosses the y-axis at the point \( (0, 4) \). Understanding this form makes graphing and understanding lines much more straightforward.

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