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Chapter 5: Problem 5

Give the slope of a line perpendicular to the given line. $$ y=x-3 $$

Short Answer

Expert verified
The slope of the line perpendicular to y=x-3 is -1.

Step by step solution

01

Identifying the Slope of the Given Line

From the form of the equation, y=mx+c, we can determine that the slope (m) of the given line is 1.
02

Determining the Slope of the Perpendicular Line

The slope (m') of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. Therefore, \( m' = -1/m = -1/1 = -1 \).

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Key Concepts

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

Understanding the Slope
The concept of "slope" is fundamental in understanding linear equations and their graphical representation. The slope of a line is a measure of its steepness and direction. It is often represented by the letter "m". In the equation of a line, which is written in the form \( y = mx + c \), the slope \( m \) tells us how much \( y \) changes for a unit change in \( x \).
  • If the slope is positive, the line ascends as it moves from left to right on the graph.
  • Conversely, if the slope is negative, the line descends.
  • A slope of zero indicates a horizontal line, while an undefined slope refers to a vertical line.
Recognizing the slope of a line is crucial when discussing perpendicular lines, as it directly influences their relationship on a graph.
Grasping Negative Reciprocals
Knowing what a "negative reciprocal" is helps in finding the slope of lines that are perpendicular to each other. A negative reciprocal essentially flips a number and changes its sign, providing a handy tool in geometry and algebra tasks.
  • If a line has a slope \( m \), the negative reciprocal of this slope is \( -1/m \).
  • This transformation means if the original slope is positive, the perpendicular slope will be negative, and vice versa.
Thus, for the example where the slope of a line is 1, the slope of a line perpendicular to it would be \(-1\). This consistent method helps in ensuring perpendicularity in linear relationships.
Exploring Linear Equations
"Linear equations" are equations that graph to straight lines and are often written in the form \( y = mx + c \). These equations are foundational in algebra and coordinate geometry, embodying relationships between two variables.
  • They provide an expressive way to depict how one quantity changes with respect to another.
  • This form lets us identify both the slope and the y-intercept ("c"), which is the point where the line crosses the y-axis.
Understanding how to manipulate and interpret linear equations is key in tasks like finding perpendicular lines, investigation of slope behavior, and solving real-life problems involving linear relationships.

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