Fundamental Ideas
Coordinate Geometry
The
slope of a line is a measurement of the steepness and direction of a nonvertical line. When a line rises from left to right, the slope is a positive number. Figure
1 (a) shows a line with a positive slope. When a line falls from left to right, the slope is a negative number. Figure
1 (b) shows a line with a negative slope. The
x-axis or any line parallel to the
x-axis has a slope of zero. Figure
1 (c) shows a line whose slope is zero. The
y-axis or any line parallel to the
y-axis has no defined slope. Figure
1 (d) shows a line with an undefined slope.
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If
m represents the slope of a line and
A and
B are points with coordinates (
xl,
y1) and (
x2,
y2) respectively, then the slope of the line passing through
A and
B is given by the following formula.
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A and B cannot be points on a vertical line, so x1 and x2 cannot be equal to one another. lf x1 = x2, then the line is vertical and the slope is undefined.
Example 1: Use Figure
2 to find the slopes of lines
a, b, c, and
d.
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(a) Line a passes through the points (−7, 2) and (−3, 4).
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(b) Line b passes through the points (2, 4) and (6, −2).
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(c) Line c is parallel to the x-axis. Therefore, m = 0.
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(d) 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 of 2/3. If another point on this line has coordinates (
x, 12), find
x.
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