# 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:

`m`

is the slope`c`

is 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 `(x`

and _{1}, y_{1})`(x`

. The slope between the points is:_{2}, y_{2})

`slope = (y`_{2} - y_{1}) / (x_{2} - x_{1})

Thus, the equation is:

`y - y`_{1} = (y_{2} - y_{1}) / (x_{2} - x_{1}) * (x - x_{1})

**Also Check: Cosine Function**

### Point-Slope Form

Given a point `(x`

and slope _{1}, y_{1})`m`

, the equation is:

`y - y`_{1} = m (x - x_{1})

**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.