What is Equation of a Line? How to Find Equation of a Line Using Slope with Examples? - BYJUS

# Equation Of A Line

A line is a one-dimensional figure formed by a series of points. We can describe a line with an equation if we know any two points that lie on the line. The steepness of a line is defined as the slope of the line. Learn the various methods of framing an equation for a line with the help of some solved examples....Read MoreRead Less

## What is a line?

A one-dimensional figure with length but no width is known as a line. A line is made up of a series of points that are infinitely extended in opposite directions. Two points in a two-dimensional plane determine a line. Collinear points are two points that are located on the same line.

## Slope of a line

The slope of a line indicates the line’s steepness and direction.

Any two distinct points on a line can be used to calculate the slope of the line. The slope of a line formula determines the ratio of “vertical change” to “horizontal change” between two points on a line.

In mathematics, the slope of a line is the change in the $$y-$$coordinate for a change in the $$x-$$coordinate.

The net change in the $$y-$$coordinate is denoted by $$\Delta y$$ and the net change in the $$x-$$ coordinate is denoted by $$\Delta x$$. So, the change in the y-coordinate for a change in the $$x-$$ coordinate is expressed as

m = $$\frac{Change~ in~ y~ coordinates}{Change~ in~ x~ coordinates}$$= $$\frac{\Delta y}{\Delta x}$$

The letter $$‘m’$$ is commonly used to symbolize the slope.

## Point slope form

In point-slope form, a linear equation written as $$y-y_1=m(x-x_1)$$. The slope of the line is , and it passes through the point $$(x_1,y_1)$$  .

Equation: $$y-y_1=m(x-x_1)$$

## Point slope form derivation

Assume $$P_1(x_1, y_1)$$ is a fixed point on a non-vertical line with m as its slope.

Consider $$P(x,y)$$ to be any point on the line L.

The ratio of the difference of the coordinates to the difference of x-coordinates is the slope of the line passing through the points $$(x_1, y_1)$$ $$(x, y)$$

$$m= \frac{(y-y_1)}{(x-x_1)}$$

$$\Rightarrow~y-y_1=m(x-x_1) \ldots(i)$$

As a point, $$p_1(x_1,y_1)$$ , which includes all points $$(x,y)$$ on L, satisfies the equation I and no other point in the plane does.

As a result, the equation I will be without a doubt the equation for the given line L.

Thus, if and only if, the point $$(x,y)$$ lies on the line with slope m passing through the fixed point $$(x_1,y_1)$$ its coordinates satisfy the equation.

$$y-y_1=m(x-x_1)$$

As a result, this is the line equation’s point-slope form.

## What are the steps to solve the point slope form of a line?

We can follow the steps below to solve point slope form for a given straight line and find the equation of the given line.

Step 1: Write down the slope of the straight line,  ‘m’, and the coordinates  $$(x_1, y_1)$$ of a given point on the line.

Step 2: In the point slope formula, substitute the given values:  $$y-y_1=m(x-x_1)$$ .

Step 3: Simplify to get the line equation in the standard form.

## Solved Equation of a Line Examples

Example 1:

Write an equation for a line that passes through the point (-2,1) and has a slope of $$\frac{2}{3}$$ in the point-slope form.

Solution:

According to the point-slope formula we know that,

Equation of line is:

$$y-y_1= m(x-x_1)$$   Write the point slope form

$$y-1=\frac{2}{3}\left [ x-(-2) \right ]$$   Substitute $$\dfrac{2}{3}$$  for m, -2 for $$x_1$$, and 1 for $$y_1$$.

$$y-1=\dfrac{2}{3}(x+2)$$         Simplify

So,  $$y-1=\dfrac{2}{3}(x+2)$$ is the equation.

Example 2:

Write an equation for the line that passes through the point (-4,2) and has a slope of $$\frac{1}{3}$$ in the point-slope form.

Solution:

According to the point-slope formula we know that,

Equation of line is:

$$y-y_1=m(x-x_1)$$   Write the point slope form

$$y-2 =\dfrac{1}{3}\left [ x-(-4) \right ]$$   Substitute $$\dfrac{1}{3}$$  for m, -4 for $$x_1$$, and 2 for $$y_1$$

$$y-2=\frac{1}{3}(x+4)$$         Simplify

So,  $$y-2=\frac{1}{3}(x+4)$$ is the equation.

Example 3:

Write an equation for the line that passes through the point (1,2) and has a slope of $$\frac{3}{5}$$ in the point-slope form.

Solution:

According to the point-slope formula we know that,

Equation of line is $$y-y_1=m(x-x_1)$$

$$y-2=\dfrac{3}{5}\left [ x-(1) \right ]$$   Substitute$$\dfrac{3}{5}$$  for m, 1 for$$x_1$$, and 2 for $$y_1$$

$$y-2=\dfrac{3}{5}(x-1)$$     Simplify

So, $$y-2=\dfrac{3}{5}(x-1)$$ is the equation.

Frequently Asked Questions on Equation of a Line:

$$y-y_1=m(x-x_1)$$ is the equation of a line passing through the point $$(x_1, y_1)$$ with slope $$m$$.

This is the equation for the point slope form.

The formula for the point-slope form of an equation is used to find the equation of a straight line given a point on it and the slope of the line.