Find the slope of each line, provided that it has a slope. through (8, 7) and (1/2, -2)

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Recall that the slope $ m $ of a line passing through two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Identify the coordinates of the two points: $ (x_1, y_1) = (8, 7) $ and $ (x_2, y_2) = \left( \frac{1}{2}, -2 \right) $.

Substitute the values into the slope formula: \[ m = \frac{-2 - 7}{\frac{1}{2} - 8} \]

Simplify the numerator and denominator separately: - Numerator: $ -2 - 7 = -9 $ - Denominator: $ \frac{1}{2} - 8 = \frac{1}{2} - \frac{16}{2} = -\frac{15}{2} $

Rewrite the slope expression with the simplified numerator and denominator: \[ m = \frac{-9}{-\frac{15}{2}} \] From here, you can proceed to simplify the complex fraction to find the slope.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Slope of a Line

The slope of a line measures its steepness and direction, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. It is given by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are coordinates of the points.

Coordinate Points

Coordinate points represent specific locations on the Cartesian plane, expressed as (x, y). Understanding how to identify and use these points is essential for calculating slope, as the differences in their x and y values determine the line's steepness.

Undefined Slope

A line has an undefined slope when the change in x (run) is zero, meaning the line is vertical. In such cases, the slope formula results in division by zero, indicating no defined slope value. Recognizing this helps avoid calculation errors.

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