Problem 93

Question

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

Step-by-Step Solution

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Answer

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.

1Step 1: 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.

2Step 2: 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 $m = (y_2 - y_1) / (x_2 - x_1)$.

3Step 3: 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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