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Chapter 2: Problem 93

What is the slope of a line and how is it found?

Short Answer

Expert verified
The slope of a line is defined as the vertical change (rise) for each unit of horizontal change (run), and can be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\), where (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)) are two points on the line.

Step by step solution

01

Definition of Slope

The slope of a line is a numerical measure of the steepness of that line. It measures the vertical change (rise) for each unit of horizontal change (run). In a two-dimensional Cartesian coordinate system, the slope of a line can be calculated if two points on that line are known.
02

Slope Calculation using Rise-over-Run Concept

To calculate the slope of a line, the concept of 'rise over run' is used. This concept refers to the amount the y-coordinate increases or decreases (the rise) for each unit that the x-coordinate increases or decreases (the run). if \(y_2\) and \(y_1\) are y-coordinates of two points on the line, and \(x_2\) and \(x_1\) are the corresponding x-coordinates, then the slope \(m\) can be calculated as \(m = (y_2 - y_1) / (x_2 - x_1)\).
03

Interpretation of Slope Value

The calculated \(m\) value will give the slope of the line. This \(m\) value can be negative, positive, zero, or undefined. A positive slope means the line slants up from left to right. A negative slope means the line slants down from left to right. Zero slope means the line is horizontal and undefined slope indicates a vertical line.

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Most popular questions from this chapter

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ -x^{2}-2 x+1=0 $$

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x-2)^{2}+(y-3)^{2}=16 $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \((x-2)^{2}+(y+1)^{2}=16\) is my graph of \(x^{2}+y^{2}=16\) translated two units right and one unit down.

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