Different Forms Of The Equation Of Line
A line is a set of points that goes on forever in both directions. In mathematics, we can represent a line using an equation.
In this article, we'll explore the different forms of the equation of a line and how to work with them. We'll look at the key components and learn how to use them to describe and analyze lines.
Introduction to Equation Of Line
Lines are a big part of our everyday life, appearing in many different forms around us. By definition, a line is a collection of points that goes on forever in both directions.
In geometry, a line extends in a straight, unbroken path with no curves and has no width or thickness.
A line is considered straight if it extends in a straight path without any bends. A section of a line with two endpoints is called a line segment.
The mathematical way to describe a line is called the “Equation of a Line.” It's written as y = mx + c, where m is the slope and c is the y-intercept.
The slope of a line shows how steep the line is, and the y-intercept is where the line crosses the y-axis.
Also Check: Difference Between Variance and Standard Deviation
Different Forms of the Equation of a Line
There are several ways to write the equation of a line:
- Normal Form
- Intercept Form
- Slope-Intercept Form
- Two-Point Form
- Point-Slope Form
Normal Form
Also known as Parametric Normal Form or Perpendicular Distance Form, this line is perpendicular to another line passing through the origin. It can be derived from the slope-intercept form or point-slope form.
Example: For a line AB perpendicular to the line passing through the origin (line C),
- Coordinates of B:
(C cos θ, C sin θ) - Slope of line AB:
tan θ - Slope of line C:
-cot θ = cos θ / sin θ
So, the equation of line C is:
y - C sin θ = - (cos θ / sin θ) (x - C cos θ)
x cos θ + y sin θ = C
Intercept Form
For a line with x-intercept p and y-intercept q, touching the x-axis at (p,0) and y-axis at (0,q), the equation is:
x/p + y/q = 1
Example: If p = 5 and q = 6, then:
x/5 + y/6 = 1
6x + 5y = 30
Also Check: Convex Polygon
Slope-Intercept Form
Represented by y = mx + c, where:
mis the slopecis the y-intercept
Example: For a line with slope m and y-intercept p, the equation is:
y = mx + p
Two-Point Form
This form finds the equation of a line given two points (x1, y1) and (x2, y2). The slope between the points is:
slope = (y2 - y1) / (x2 - x1)
Thus, the equation is:
y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)
Also Check: Cosine Function
Point-Slope Form
Given a point (x1, y1) and slope m, the equation is:
y - y1 = m (x - x1)
Example: For a line with slope m = 3 passing through (4, 5), the equation is:
y - 5 = 3 (x - 4)
y = 3x - 7
3x - y - 7 = 0
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Frequently Asked Questions on Different Forms Of The Equation Of Line
The equation of a line can be written in various forms, such as the slope-intercept form (y = mx + b), the point-slope form (y - y1 = m(x - x1)), the two-point form ((y - y1) = (y2 - y1)/(x2 - x1)(x - x1)), and the standard form (Ax + By + C = 0).
The general form of the equation of a line is Ax + By + C = 0, where A, B, and C are real numbers, and at least one of A or B is non-zero. This form can represent any straight line in the coordinate plane.
The three main forms of the equation of a straight line are:
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1)
- Standard form: Ax + By + C = 0
These forms allow you to represent a line using different types of information, such as slope and y-intercept, a point and slope, or the coefficients A, B, and C.
The different forms of the line equation include:
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1)
- Two-point form: (y - y1) = (y2 - y1)/(x2 - x1)(x - x1)
- Standard form: Ax + By + C = 0
- Intercept form: x/a + y/b = 1
These forms allow you to represent a line using different types of information about the line.
The two-point form of the equation of a line is (y - y1) = (y2 - y1)/(x2 - x1)(x - x1), where (x1, y1) and (x2, y2) are two known points on the line. This form allows you to write the equation of a line using the coordinates of two points on the line.