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Chapter 2: Problem 42

Find the slope of the line satisfying the given conditions. horizontal, through \((3,5)\)

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Understand the properties of a horizontal line

A horizontal line has the same y-coordinate for all points on the line. This means there is no change in the y-value as x changes.
02

Formula for slope

The formula for the slope (m) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\).
03

Substitute values considering a horizontal line

For a horizontal line going through \( (3, 5) \), pick another point like \( (4, 5) \). Substitute these points into the formula: \(\frac{5 - 5}{4 - 3} = \frac{0}{1}\).
04

Simplify the result

Since \(\frac{0}{1} = 0\), the slope of the horizontal line is 0.

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Key Concepts

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

Understanding Horizontal Lines
A horizontal line is a special kind of line where the y-coordinate remains constant for all points along the line. This means no matter how the x-coordinate changes, the y-value stays the same.
For example, in the line going through \( (3, 5) \), every point will share the y-coordinate of 5. Other points like \( (4, 5) \) or \( (7, 5) \) will also be on this horizontal line.
Horizontal lines are easy to spot on a graph because they run parallel to the x-axis. The most important thing to remember about these lines is that the slope is always zero.
Imagine riding a bike on a completely flat road鈥攖hat's a horizontal line for you!
Explaining the Slope Formula
The slope of a line measures its steepness and direction. It helps us understand how a line changes as we move along it.
The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \ \frac{y_2 - y_1}{x_2 - x_1} \.
This formula tells us how much the y-coordinate (vertical change) changes for a unit change in the x-coordinate (horizontal change). If the value is positive, the line rises as it moves right; if negative, it falls.
In the case of a horizontal line, since there is no vertical change, the numerator (top part of the fraction) becomes zero, leading to a slope of zero.
Point-Slope Calculation for Horizontal Lines
To find the slope using the point-slope calculation, an example will help.
Consider the points \( (3, 5) \) and \( (4, 5) \) on a horizontal line. Applying the slope formula: \ \frac{y_2 - y_1}{x_2 - x_1} \.
Substitute the values: \ \frac{5 - 5}{4 - 3} = \frac{0}{1} = 0 \.
Here, the y-values (5 and 5) are the same, leading to zero in the numerator. Thus, the slope of any horizontal line is always 0.
Remember, this is true regardless of which points you choose, as long as they lie on the same horizontal line.

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Most popular questions from this chapter

Find the center-radius form of the equation of a circle with center \((3,2)\) and tangent to the \(x\) -axis.

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