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Chapter 5: Problem 25

State the slope of the lines perpendicular to the graph of \(y=3 x-2\). (Lesson \(4-6\) )

Short Answer

Expert verified
The slope of lines perpendicular to the graph is \(-\frac{1}{3}\).

Step by step solution

01

Identify the Slope of the Given Line

The equation of the line is given in the slope-intercept form as \( y = 3x - 2 \). Here, the coefficient of \( x \) is the slope of the line. Therefore, the slope \( m \) of the given line is \( 3 \).
02

Understand Perpendicular Slope Relationship

For two lines to be perpendicular, the product of their slopes must be \(-1\). If \( m_1 \) is the slope of one line and \( m_2 \) is the slope of the perpendicular line, we have the relationship \( m_1 \times m_2 = -1 \).
03

Calculate the Slope of the Perpendicular Line

Using the relationship \( m_1 \times m_2 = -1 \), where \( m_1 = 3 \) is the slope of the given line, we need to find \( m_2 \):\[ 3 \times m_2 = -1 \]Dividing both sides by \( 3 \) gives:\[ m_2 = -\frac{1}{3} \].

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Key Concepts

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

Slope of a Line
The slope of a line is a measure of its steepness and direction. In the slope-intercept form of a line, which is represented as \( y = mx + b \), the variable \( m \) indicates the slope. Here, the line's slope tells us how much the \( y \)-coordinate changes for a one-unit change in the \( x \)-coordinate. It shows a constant rate of change along the line. A positive slope, like \( 3 \) in \( y = 3x - 2 \), indicates the line is rising as it moves from left to right, while a negative slope implies it is falling. A zero slope means the line is perfectly horizontal, and an undefined slope appears when the line is vertical, creating a division by zero scenario.
Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). For two lines to be perpendicular, their slopes must be negative reciprocals of each other. The mathematical relationship for this is given by \( m_1 \times m_2 = -1 \), where \( m_1 \) and \( m_2 \) are the slopes of the two lines. This rule allows us to determine that if one line has a slope of \( 3 \), the slope of any line perpendicular to it must be \(-\frac{1}{3}\) since \( 3 \times -\frac{1}{3} = -1 \). This reciprocal and inversion relationship is crucial in geometry to ensure proper orientation and angles between lines.
Slope-Intercept Form
The slope-intercept form of a linear equation is among the most common ways to express a line on a graph. It comes in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept, which is the point where the line crosses the y-axis. This form makes it easy to quickly identify a line's slope and starting point on the graph, which helps in graphing linear equations efficiently. The y-intercept \( b \) indicates that when \( x = 0 \), \( y \) will equal \( b \). For example, in \( y = 3x - 2 \), the y-intercept is \(-2\), marking the point \( (0, -2) \) where the line crosses the y-axis.

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Most popular questions from this chapter

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