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Home > News & Articles > Straight Line Formula: Definitions, Formula, Equations, Forms, and Examples
Updated on 20th December, 2023 , 6 min read
We see a lot of items every day that are perfectly straight and aligned along lines that are comparable to our own or that are parallel to other lines that are the same length or different in length but nonetheless straight. This is a scientific idea, but mathematically, every fixed line or curve on the line segment may be described as a straight line. If we add those sides, however, we would be looking at full geometric relations where adding up lines would build up diverse forms.
Also Read: ILATE Rule
A construction having just a one-dimensional form, no width, and an infinite length can be used to represent a straight line. There appear to be an endless number of points linked between the two specified places in its construction. It appears that the straight-line structure has no bends at all. It is always possible for it to be horizontal, vertical, or inclined. The angle that appears when an angle is placed on a straight line always seems to be 180 degrees.
Railway tracks, measuring tools, and even portions of mirrors can be considered straight lines, to name a few instances of straight lines.
A straight line's general equation is y = mx + c, where m is the slope of the line and c is the y-intercept. It is the most frequent form of a straight-line equation used in geometry. A straight line is a two-dimensional geometrical element that extends indefinitely on both ends.
The equation of a straight line is atwo-variable linear equation that is fulfilled by every point on the line. That is, it is a mathematical equation that expresses the relationship between the coordinate points on that straight line. It may be expressed in several ways and indicates the slope, x-intercept, and y-intercept of a line. It may also be used to find the points on a line. The equation of a straight line is often obtained using point-slope form, slope-intercept form,two-point form, standard form, and so on. The most frequent formulas for determining the equation of a straight line are listed below-
ax + by = c
y = mx + c
y - y1= m (x - x1)
The straight line formula equation changes based on the information known about the line, such as slope, intercepts, and so on. The slope of a line with two points (x1, y1) and (x2, y2) may be determined using the formula m = (y2- y1)/(x2- x1). The following table gives details about the examples of straight-line formulae-
Different Forms of Straight Lines | Equation of Straight Line |
Equation of a horizontal line with some point (a, b) on it | y = b |
Equation of a vertical line with some point (a, b) on it | x = a |
General/Standard Form of a Line | ax + by = c |
Intercept Form (Given the intercepts a and b) | x/a + y/b = 1 |
Normal Form (Given θ = the angle made by the normal with the positive direction of the x-axis and p = distance of the line from the origin) | x cos θ + y sin θ = p |
Point-Slope Form (Given slope m and point (x1, y1)) | y - y1= m (x - x1) |
Slope - Intercept Form (Given slope m and y-intercept (0, c)) | y = mx + c |
Two Point-Form (Given two points (x1, y1) and (x2, y2) on the line) | y - y1= (y2- y1)/(x2- x1) (x - x1) |
A straight line's equation often includes a slope. If a line creates an angle with the positive direction of the x-axis, then the angle is known as the line's inclination, and the slope is known as the line's slope. Take note of thex-axis's slope of 0. In actuality, the slope of all lines parallel to the x-axis is 0. Additionally, the slope of every line parallel to the y-axis, including the y-axis itself, is not known. The following are some of the forms of the straight-line formula-
The point-slope form is used to get the equation of a straight line through the point (x1, y1) with slope m. The point-slope form's equation is-
Where (x, y) is any point on the line, y - y1= m (x - x1).
We are aware of the following equation for a line's slope-
⇒ m = (y - y1)/(x - x1)
Multiplying both sides by (x - x1),
m (x - x1) = (y - y1)
This can be written as,
(y - y1) = m (x - x1)
Thus, the proof for the point-slope form of the equation of a straight line.
A line with the two points (x1, y1) and (x2, y2) on it is an example. The slope's value may then be determined using the formula m = (y2- y1)/(x2 - x1). When this is substituted in the point-slope form mentioned above, the two-point form is obtained as y - y1= (y2- y1)/(x2- x1) (x - x1).
Now imagine that you are given a line with the slope m and the y-intercept. Let's say a line crosses the y-axis at (0, c). We may write y - c = m (x - 0) y = mx + c, where c is the y-intercept, using the point-slope form. This is referred to as a line's slope-intercept form.
NOTE: The slope-intercept version of the line's equation is y = m (x - d) if d is the x-intercept.
If a line's x- and y-intercepts are (a, 0) and (0, b), then, correspondingly. The slope of the line is thus m = (b - 0)/(0 - a) = -b /a. The point-slope form of its equation is thus as follows-
y - 0 = -b/a (x - a)
Multiplying both sides by a,
ay = -bx + ab
bx + ay = ab
Dividing both sides by ab,
x/a + y/b = 1
A straight line's standard form is provided by the equation axe + by = c, where a, b, and c are real values. Any configuration of a line can be considered standard. Let's look at an illustration of how to put the equation y = 2x - 1 in standard form. Add 2x to neither side of the equation, and we get
y - 2x = 2x - 1 - 2x
⇒ y - 2x = -1
⇒ 2x - y = 1
As a result, we arrive at the line's conventional form, which is 2x - y = 1.
The following are some of the steps to follow to find the line equation-
Step 1: Write down the given information, the line's slope as "m," and the coordinates of the given point or points in the form (xn, yn).
Step 2: Based on the provided parameters, apply the necessary formula.
Step 3: Rearrange the terms to get a standard form equation for the line.
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By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readAns. Simply by simplifying the equation such that the left side of the equation has only y and all other components are shifted to the right side, a straight-line equation may be changed from the point-slope form to the slope-intercept form. For instance, the point-slope form of y - 2 = 3 (x - 1) can be used. Simplifying it, we get y - 2 = 3x - 3 and by adding 2 to both sides, we get y = 3x - 1, or in slope-intercept form.
Ans. It is only possible to find an equation in normal form, which reads x cos + y sin = p when the normal from the origin to a line forms an angle with the positive direction of the x-axis.
Ans. A straight line equation is a linear equation with variables x and y that describes the relationship between the coordinate locations along the line. A straight line’s equation often has the form y = mx + c, where m denotes the line’s slope and c its y-intercept.
Ans. An x-intercept is a mathematical term used to describe the area where a line seems to cross the x-axis. The definition of the y-intercept is completely at odds with this. The intercepts aid in creating correct linear equations.
Ans. We may get the slope m and the y-intercept c of a straight-line equation by changing it to the slope-intercept form y = mx + c. For instance, if the equation is 2x - 3y = 1, we must first solve it for y before we can determine its slope and y-intercept. Y is thus equal to (2/3)x - 1/3. When we compare this to the equation y = mx + c, we obtain the slope m = 2/3 and the y-intercept (0, c) = (0, -1/3).
Ans. A given place may always be represented by two integers on the cartesian plane. These dots on the plane that appear to be coordinates appear to indicate different numbers.
Ans. The y-intercept, also known as the vertical intercept, seems to be a point that denotes the intersection of a certain function; the intersection point must be on an x-y cartesian plane.
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