Slope of a Line

The slope of a line is a measurement of the steepness and direction of a nonvertical line. When a line slants from lower left to upper right, the slope is a positive number. Item (a) in Figure 1 shows a line with a positive slope. When a line slants from lower right to upper left, the slope is a negative number (b). The x- axis or any line parallel to the x-axis has a slope of zero; that is, a horizontal line has a slope of zero (c). The y-axis or any line parallel to the y-axis has no defined slope; that is, a vertical line has an undefined slope (d).

Figure 1 Slopes of lines.

If m represents the slope of a line and A and B are points with coordinates ( x₁, y₁) and ( x₂, y₂), respectively, then the slope of the line passing through A and B is given by the following formula.

Since A and B cannot be points on a vertical line, x₁ and x₂ cannot be equal to one another. If x₁ = x₂, then the line is vertical, and the slope is undefined.

Example 1: Use Figure 2 to find the slopes of the lines a, b, c, and d.

Figure 2 Find the slopes.

Line a passes through the points (−7,2) and (−3,4).

Line b passes through the points (2,4) and (6,−2).

Line c is parallel to the x-axis. Therefore,

m = 0

Line d is parallel to the y-axis. Therefore, line d has an undefined slope.

Example 2: A line passes through (−5,8) with a slope of⅔. If another point on this line has coordinates ( x,12), find x.

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