A * line* is a one dimensional representation which has no thickness and which can be extended to infinite length.

Example: - 3x + 5y = 2 is an example of a line in the standard form.

There will be no fractions in the standard form of a line.

Lets consider a few examples

Find the equation of a line passing through the point (3, 4) with slope 2 in the standard form.

Given
that the required line is passing through the point (3, 4).

Slope of the line = 2.

First lets write the line using point slope form.

We know that when a point (x_{1}, y_{1}) and slope m is given, then the point slope form is

y - y_{1} = m (x - x_{1}).

Substituting the slope and points we get

y - 4 = 2 (x - 3)

Expanding we get

y - 4 = 2x - 6

Now we to take all
variables to one side, keeping the constants on the other side.

Adding 4 on both sides, we get

y - 4 = 2x - 6

__ +4 + 4__

y = 2x - 2

Now we subtract 2x from both sides

y = 2x - 2__ -2x -2x __

y -2x = -2**So the required equation is -2x + y = -2**

**Example 2: -**Find the slope of the equation 3x - 4y = 9.

The given equation is in the standard form.

We know that the formula for finding slope is given by

Here coefficient of x = 3 and coefficient of y = -4

So slope of the equation = -(3/(-4)) =

**Example 3: -**

Find the equation of the line passing through the points (4, 2) and (5, 3) and hence find the slope.**Solution: -**

Given
that the required line is passing through the points (4, 2) and (5,3).

First lets write the line using two point form.

We know that when two points (x_{1}, y_{1}) and (x_{2}, y_{2}) are given, then the equation of the line is is

y - y_{1} = m (x - x_{1}) where m = (y_{2} - y_{1})/(x_{2} - x_{1})

Here (x_{1}, y_{1}) = (4, 2) and (x_{2}, y_{2}) = (5, 3)

So m = (y_{2} - y_{1})/(x_{2} - x_{1}) = (3 - 2)/(5 - 4) = 1/1 = 1

Substituting the slope and points we get

y - 2 = 1 (x - 4)

Expanding we get

y - 2 = x - 4

Now we to take all
variables to one side, keeping the constants on the other side.

Adding 2 on both sides, we get

y - 2 = x - 4

__ +2 + 2__

Now we subtract x from both sides

y = x - 2

y -x = -2

Here coefficient of x = -1 and coefficient of y = 1

So slope of the equation = -(-1/(1)) =

**Example 4: -**

Find the equation of the line passing through the points (-3, 4) and (-7, 9) and hence find the slope.**Solution: -**

Given
that the required line is passing through the points (-3, 4) and (-7, 9).

First lets write the line using two point form.

We know that when two points (x_{1}, y_{1}) and (x_{2}, y_{2}) are given, then the equation of the line is is

y - y_{1} = m (x - x_{1}) where m = (y_{2} - y_{1})/(x_{2} - x_{1})

Here (x_{1}, y_{1}) = (-3, 4) and (x_{2}, y_{2}) = (-7, 9)

So m = (y_{2} - y_{1})/(x_{2} - x_{1}) = (9 - 4)/(-7 - (-3)) = 5/-4 = -5/4

Substituting the slope and points we get

y - 4 = -5/4 (x + 3)

Expanding we get

y - 4 = -5/4x - 15/4

Here we have a fraction in the equation. So we multiply the whole equation by the denominator 4.

We get

4y - 16 = -5x -15.

Now we to take all
variables to one side, keeping the constants on the other side.

Adding 16 on both sides, we get

4y - 16 = -5x - 15

__ +16 + 16__

Now we subtract -5x from both sides. That is add 5x on both sides

4y = -5x + 1

4y + 5x = 1

Here coefficient of x = 5 and coefficient of y = 4

So slope of the equation = -(5/(4)) =

**Try Yourself: -**

A line is passing through (2,3) and (4,2). Write the equation in the standard form. Also find the slope of the line.

Equation x + 2y = 8 slope = -1/2