# Linear Equations In Two Variables

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 â€˜1â€™ 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 aâ‰ 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,

$2x + 4$ = $0$,

â‡’ $x$ = $-\frac{4}{2}$

â‡’$x$ = $-2$

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 â€˜Aâ€™ and â€˜Bâ€™ is 32. How is it represented in terms of an equation?

Let x and y be the number of goals scored by the players â€˜Aâ€™ and â€˜Bâ€™ respectively.

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

Therefore,

$x~ +~ y$ = $32$ Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â — -(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.

$âˆš2~t~ +~ 4v$ = $-5$, 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 aâ‰ 0 can also be a linear equation in two variables because, it can be represented as,

$ax ~+~ 0~Ã—~y ~+ ~b$ = $0$

For example;

$2x~ +~ 5$ = $0$ can be written as, $2x~ +~ 0~ Ã—~ y ~+ ~5$ = $0$.

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

1. âˆš3 x + y = 5

It can be written as, âˆš3 x + y – 5 = 0

Therefore,a=âˆš3, b=1 and c=-5

2 . 2y + 3 = 2

It can be written as, 0 Ã— 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

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Thus, Â linear equations in two variables are an important part of mathemics domain. Â 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.

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A linear equation in two variables