Equation of a Line Through Point (2, -2) with Various Slopes
The equation of a line is crucial in various fields, from mathematics to engineering. Given a specific point and a slope, we can formulate the equation of the line that passes through that point. This article discusses how to derive the equation of a line that passes through the point (2, -2) with different slopes. We'll explore three different scenarios based on the given information.
Determining the Slope
The slope (m) of a line is a measure of its steepness. It describes how much the line rises or falls for a given horizontal run. In mathematical terms, if we have two points (x1, y1) and (x2, y2) on a line, the slope is given by:
[ m frac{y2 - y1}{x2 - x1} ]
In the given problem, we have the point (2, -2). Let's examine the three possible scenarios for the slope (m) and derive the corresponding line equations.
Slope as an Expression (m-21)
If the slope is given as an expression 'm-21', we treat it as a single entity. The equation of the line in point-slope form is:
[ y - y1 m(x - x1) ]
Substituting the point (2, -2) and the slope 'm-21', we get:
[ y - (-2) (m-21)(x - 2) ]
Expanding and simplifying:
[ y 2 (m-21)(x - 2) ]
[ y 2 mx - 2m - 21x 42 ]
[ y mx - 21x - 2m 42 - 2 ]
[ y mx - 21x - 2m 40 ]
Slope as a Specific Value (-21)
If the slope is given as -21, we directly use this value. The equation of the line in point-slope form is:
[ y - y1 m(x - x1) ]
Substituting the point (2, -2) and the slope -21, we get:
[ y - (-2) -21(x - 2) ]
Expanding and simplifying:
[ y 2 -21(x - 2) ]
[ y 2 -21x 42 ]
[ y -21x 42 - 2 ]
[ y -21x 40 ]
Slope as a Positive Value (21)
If the slope is given as 21 and the minus sign is just a connector, then the slope is 21. The equation of the line in point-slope form is:
[ y - y1 m(x - x1) ]
Substituting the point (2, -2) and the slope 21, we get:
[ y - (-2) 21(x - 2) ]
Expanding and simplifying:
[ y 2 21(x - 2) ]
[ y 2 21x - 42 ]
[ y 21x - 42 - 2 ]
[ y 21x - 44 ]
Summary
In conclusion, the equation of a line passing through the point (2, -2) with the slopes discussed can be summarized as:
If the slope is 'm-21', then the equation is (y mx - 21x - 2m 40). If the slope is -21, then the equation is (y -21x 40). If the slope is 21, then the equation is (y 21x - 44).Understanding how to manipulate these equations and apply the point-slope form can be incredibly useful in a variety of mathematical and real-world applications.
Related Keywords
line equation, slope, point-slope form