Problem 5 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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, \( m' = -1/m = -1/1 = -1 \).

Key Concepts

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.