Calculating the slope of a line is a fundamental concept in algebra, often summarized as "rise over run." The slope formula can be expressed as:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \( m \) represents the slope, \( y_2 \) and \( y_1 \) are the y-coordinates of two points on the line, and \( x_2 \) and \( x_1 \) are the corresponding x-coordinates. The slope indicates how steep a line is and the direction it moves. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.
To calculate the slope from a graph, start by identifying two points where the line intersects the grid. For example, if you select points at coordinates (1, 5) and (2, 4), you would first determine the rise:
Rise = \( y_2 - y_1 = 4 - 5 = -1 \)
Next, calculate the run:
Run = \( x_2 - x_1 = 2 - 1 = 1 \)
Now, substitute these values into the slope formula:
\( m = \frac{-1}{1} = -1 \)
This indicates a slope of -1, meaning the line descends as it moves to the right.
In another example, if you choose points (3, 2) and (6, 3), the rise would be:
Rise = \( 3 - 2 = 1 \)
And the run would be:
Run = \( 6 - 3 = 3 \)
Thus, the slope is:
\( m = \frac{1}{3} \)
In this case, the line has a gentle upward slope.
For a third example, selecting points (3, 4) and (4, 6) gives a rise of:
Rise = \( 6 - 4 = 2 \)
And a run of:
Run = \( 4 - 3 = 1 \)
Calculating the slope yields:
\( m = \frac{2}{1} = 2 \)
This indicates a steep upward slope. The comparison of slopes reveals that a higher slope value corresponds to a steeper line. For instance, a slope of 2 is significantly steeper than a slope of \( \frac{1}{3} \).
Understanding slope is crucial as it not only describes the steepness of a line but also provides insights into the relationship between variables in various contexts, such as in physics, economics, and statistics.
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