Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (4, -1) and (3, −1)

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Identify the coordinates of the two points: Point 1 is (4, -1) and Point 2 is (3, -1).

Recall the formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\): \(slope = \frac{y_2 - y_1}{x_2 - x_1}\)

Substitute the given points into the slope formula: \(slope = \frac{-1 - (-1)}{3 - 4}\)

Simplify the numerator and denominator separately: Numerator: \(-1 - (-1) = -1 + 1 = 0\) Denominator: \(3 - 4 = -1\)

Calculate the slope using the simplified values: \(slope = \frac{0}{-1}\) which equals 0, indicating the line is horizontal and does not rise or fall.

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주요 개념

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Slope of a Line

The slope measures the steepness of a line and is calculated as the change in y-values divided by the change in x-values between two points. It is given by the formula m = (y2 - y1) / (x2 - x1). This value indicates how much y changes for a unit change in x.

Undefined Slope

A slope is undefined when the denominator in the slope formula (x2 - x1) equals zero, meaning the line is vertical. Vertical lines have no horizontal change, so their slope cannot be expressed as a real number.

Interpreting Slope Direction

The sign of the slope indicates the line's direction: a positive slope means the line rises from left to right, a negative slope means it falls, zero slope means the line is horizontal, and an undefined slope means the line is vertical.