Before knowing Linear equations in two variables, let us know what are linear equations in one variable.

**Linear equations in one variable**

Equations having degree ‘1’ is known as linear equations. Standard form of a linear equation in one variable is, ax+b=0, where a and b are real numbers and a≠0.

While solving a linear equation,

- Same number can be added to (or subtracted from) both sides of the equation.
- Both sides of the equation can be multiplied or divided by same non- zero number.

For example, \(2x~ + ~4\) = \(0\) is a linear equation in variable x.

The solution or root of the equation is,

\(2x + 4\) = \(0\),

⇒ \(x\) = \(-\frac{4}{2}\)

⇒\(x\) = \(-2\)

It is represented on the number line as follows,

**Linear equations in two variables**

Consider the situation,

In a football tournament, sum of the goals scored by two players ‘A’ and ‘B’ is 32. How is it represented in terms of an equation?

Let x and y be the number of goals scored by the players ‘A’ and ‘B’ respectively.

It is given that total number of goals scored by them is 32.

Therefore,

\(x~ +~ y\) = \(32\) — -(1)

The above equation is an example of linear equation in two variables, where x and y are the two variables. It is not compulsory that the two variables can only be written as x and y. It is customary to denote variables in terms of x and y in these equations.

Few more examples for linear equation in two variables are given below,

\(5p ~+~ 4q\) = \(12\), where p and q are the two variables.

\(2.3u ~+~ 1.8v~ -~ 5\) = \(0\), where u and v are the two variables.

\(√2~t~ +~ 4v\) = \(-5\), where t and u are the two variables.

Therefore, general form of a linear equation in two variables is

\(ax~ +~ by ~+ ~c\) = \(0\)

where a, b and c are real numbers, and both a and b are not equal to zero.

An equation of the form \(ax~+~b\) = \(0\), where a and be are real numbers, and a≠0 can also be a linear equation in two variables because, it can be represented as,

\(ax ~+~ 0~×~y ~+ ~b\) = \(0\)

For example;

\(2x~ +~ 5\) = \(0\) can be written as, \(2x~ +~ 0~ ×~ y ~+ ~5\) = \(0\).

Example: Write the equations given in the form of \(ax~+~by~+~c\) = \(0\) and find the values of a, b and c.

1. √3 x + y = 5

It can be written as, √3 x + y – 5 = 0

Therefore,a=√3, b=1 and c=-5

2 . 2y + 3 = 2

It can be written as, 0 × x + 2y + 1 = 0

Therefore,a=0, b=2 and c=1

3. 4y + 2x – 2 = 0

It can be written as, 2x + 4y – 2 = 0

Therefore,a=2, b=4 and c=-2

Thus, linear equations in two variables are an important part of mathemics domain. To understand this topic, students are expected to practice with sample questions and answers like the ones provided here- NCERT solutions for linear equations in two variables.