General Equation of a Line

In simple terms, we can say the line is a figure which has no curvature. In two variables, the general equation of a line of the first degree is represented as

Ax + By +C = 0,

A, B ≠ 0 where A, B, and C are constants that belong to real numbers.

Coordinate of a point is denoted by writing the x-coordinate first and then y-coordinate, which is separated by a comma.

What is a Line?

A line is defined by its length but not by its width. A line could be a  two-dimensional geometric figure which will move in any direction. There are an infinite number of points that conjure a line. On all sides, it's endless and nonstop. A line could be a one-dimensional object. Ancient mathematicians established the concept of line or line in geometry to depict straight objects with negligible breadth and depth. It's frequently explained in terms of two points.

Different Types of Lines

In Geometry, there are various varieties of lines. Geometry is made on the inspiration of lines. Horizontal lines (sleeping lines), Vertical lines (sleeping lines), and Oblique lines are the three kinds of straight lines (Slanting lines).

1. A horizontal line could be a line that runs parallel to the x-axis. The x-axis is paralleled by a horizontal line. This line doesn't cross the X-axis at any point.

2. A vertical line is one that's parallel to the y-axis. It runs straight up and down, parallel to the coordinate plane's y-axis.

3. An oblique line is the one which is neither horizontal nor vertical.

Parallel lines are defined as two lines that run parallel and are separated by the same distance but don't meet, no matter their length. Parallel lines are those that don't touch or intersect and maintain a relentless minimum distance between them. Parallel lines are two straight lines in a very plane that don't meet at any point.

Intersecting lines are formed when two non-parallel lines intersect at a point. Two lines that intersect at the identical location are called intersecting lines.

A straight line's general equation is y = mx + c, where m is the line's slope and c is the y-intercept. In geometry, it's the foremost common type of equation of a line. A straight line's equation may be stated in an exceeding style of ways, including point-slope form, slope-intercept form, general form, standard form, and so on. A line may be a geometrical entity with two dimensions that continues indefinitely on both ends.

A line equation could be a mathematical equation that expresses the link between the coordinate points on a line. It is expressed in a sort of way and indicates the line's slope, x-intercept, and y-intercept. The foremost popular variants of the line equation are y = mx + c and ax + by = c. Point-slope form, slope-intercept form, general form, and standard form are samples of other forms.

A line could be a figure constructed by connecting two points A (x₁ , y₁) and B (x₂ , y₂) with the shortest distance between them and increasing both ends to infinity.

Linear equations are written in three different ways: point-slope form, standard form, and slope-intercept form.

1. The slope of a line is denoted by m, and also the y-intercept is denoted by c. When B = 0, the classic first-degree equation Ax + By + C = 0 is expressed as a slope-intercept equation.

2. The purpose where a line crosses the x-axis or the y-axis is named the intercept. Assume that a line intersects the x- and y-axes at (a, 0) and (0, b), respectively. ON  the x-axis and also the y-axis, the equation of a line intercepts.

3. The angle formed by the perpendicular with the positive x-axis and also the equation of the line whose length of the perpendicular from the origin is p is named Normal Line.

Slope (m) of a non-vertical line passing through the points (x₁ , y₁ ) and (x₂ , y₂)

m=(y₂ - y₁)/(x₂ - x₁), x₁ ≠ x₂

Equation of a horizontal line

y = a or y = -a

Equation of a vertical line

x = b or x = -b

Equation of the line passing through the points (x₁ , y₁) and (x₂ , y₂)

y - y₁= [(y₂ - y₁)/(x₂ - x₁)]×(x - x₁)

Equation of line with slope m and intercept c

y = mx+c

Equation of line with slope m makes x-intercept d.

y = m (x – d).

Intercept sort of the equation of a line

(x/a)+(y/b)=1

The normal sort of the equation of a line

x cos α+y sin α = p

How to Find the Equation of a Straight Line that is Parallel to One of the Coordinate Axes.

To find the normal form of a straight line, consider we have two axes in a two-dimensional form that is x and y.

A straight line AB cuts the x-axis at point A. This line AB is parallel to axis y.

We can consider the distance OA = a. Let P be any point on line AB such that the coordinates of P are (x, y).

The abscissa of the point is always c.

       x = c

This is true for every point on the line AB.

It should be noted that the equation does not contain the coordinate y.

Similarly, we can find the equation of a straight line parallel to the x-axis.

      y = d

The equation of the x-axis is y = 0

The equation of y-axis is x = 0.

How to Find the Equation of a Straight that Cuts Off a Given Intercept on the Axis of Y and is Inclined at an Angle to the X-axis.

Let the intercept be c and the angle be a.

Let c be a point on the y-axis such that the distance  OC is c.

Draw a perpendicular PM to OX and a line parallel to the x-axis that is CN. See figure below

(Image will be uploaded soon)

(Image will be uploaded soon)

Let the coordinates of P be (x, y) such that OM = x and PM = y.

MP = NP + MN

By trigonometry, we get NP = CN. tan(a)

Therefore,

We know that tan a = m ( also called the slope of the line ).

So, y = CN.tan(a) + OC

= mx + c

Therefore, 

We get the general equation of a line which is

y = mx +c

Solved Example

Example 1: Find the equation of the straight line cutting off an intercept 3 in the negative direction of the y-axis and inclined at 120° to the axis of x. 

Answer 

y = x.tan(120°) +(-3)

y + x√3 + 3 = 0.

Example 2: How to find the equation of a line that cuts off given intercepts a and b on x-axis and y-axis respectively.

Given is that OA = a and OB = b

Now join AB and extend it in both ways making it a line. Let P be any point which has coordinates (x, y) present on this straight line. 

Draw PM perpendicular to OX and touching line AB at P as shown in fig below.

By geometry we get 

OM/OA = PB/AB and MP/OB =AP/AB

Adding LHS for both of these we get

OM/OA + MP/OB =( PB + AP)/AB 

= 1

Therefore we get 

x/a + y/b =1

How to Find the Equation of a Straight Line in Terms of Perpendicular which Falls Upon it from the Origin and the Angle Which this Perpendicular Makes with the X-axis.

Step1: Let the given line be AB (for which we need to find  its equation).

Step 2: Let OR be perpendicular to AB. Its length is let's say p.

Step 3: Let a be the angle that OR makes with OX. Let P be any point whose coordinates are x and y lying on AB. 

Step 4: Draw the ordinate PM and ML perpendicular to OR. All of these is given in the image below

OL = OM cos(a)...................(1)

LR = NP = MP sin (NMP)

Ang NMP =90° - ang NMO = ang MOL= a

LR = MP sin(a).......................(2)

By adding 1 and 2 we get

OM cos a + MP sin a = OL + LR = OR = p

Therefore, xcosa + ysina = p

In all the above forms of the line, we see that they are of first degree in x and y.

We now know that any equation of first degree in x and y always represents a straight line.

A general form of this equation is :

Ax + By + C = 0

Here A, B, C are constants ie these are quantities that don't contain x and y.

Ax + By + C = 0 it may also be written as

y = {-A/B}(x) - C/B

Comparing this above equation with y= mx + c we get

m =-A/B and c = -C/B

Therefore the equation Ax + By + C = 0 represents a straight line cutting off an intercept -C/B to the axis of y and inclined at an angleTan⁻¹(-A/B).

How to Find the Equation of a Straight Line Which Passes Through Two Given Points (-1, 3) and (4,-2)

We know that equation to any straight line is y = mx + c ;..................(1)

for first point we get

3 = -m + c, such that c = m + 3

Hence (1) becomes

y = mx + m + 3……………(2)

For the second point, we have

-2 = 4m + m +3

So solving this we get m = -1,

Hence  (2) becomes 

y = -x + 2 => x + y = 2

If two lines are parallel to each other we get their slopes to be the same.

That is m₁=m₂.

If two lines are perpendicular we get 

m₁.m₂ = -1

If two lines subtend an angle let's say "a" between them then 

Tan(a) =(m₁ - m₂)/1 + m₁m₂

To fix the position of a straight line we must always have two quantities given. Such that one is the point and the other is a quantity that gives the direction of a line.

Conclusion

Nearly every facet of our daily lives involve lines and angles. To achieve math examinations, students must be confident in calculating angles, measuring angles, and drawing angles; nevertheless, a radical understanding of lines and angles may aid students' comprehension of the globe.

Angles determine whether a building is safe or not within the construction sector. To create a structure that stands upright and allows rainfall to escape the roof, architects and constructors must calculate angles very precisely. Furthermore, construction workers cannot make sure that windows and doors will fit until all structures are created with straight lines.