**Linear equations in two variables**". In previous lessons we have already learnt basic linear equations and linear equations in one variable.

## Linear Equations in Two Variables Definition

A linear equation in two variables is an equation of the form ax + by + c = 0 or ax + by = c, where a, b,c are the constants (real numbers) and that a,b are not both zero (a ≠ 0, b ≠ 0).

**Examples:**Here are some perfect examples of linear equations in two variables,

- 2x - 3y = 5
- 3x + 4y + 7 = 0
- 2x + 3y - 6 = 0
- x + 4y = 7

Above mentioned all the examples are linear equations in two variables. Thus, there are two variables (x,y) in every linear equation.

## Forms of linear equations in two variables

The form also can be called as the formula of linear equations in two variables and it is represented as:

ax + by + c = 0 or ax + by = c

Where,

- a, b and c are the real numbers
- x and y are the variables

## Methods for Solving Linear Equations on Two Variables

There are mainly two methods that are mostly used to solve a linear equation in two variables. The methods are as follows,

- Substitution method
- Elimination method
- Cross multiplication method

### 1. Method of Substitution

**Step 1:**There must be two variables in linear equations in two variables. You need to find the value of one variable first, say x in terms of the other.

**Step 2:**Now substitute the value of x in the other equation. Hence, we get an equation of only one variable y.

**Step 3:**Now you can easily solve the equation for y. Substitute the value of y in the first equation to solve for variable x.

**Example:**Solve x + y = 5, 3x - 2y = 10

**Solution:**The given equations are

**(First step)**

x + y = 5 . . . (1)

3x - 2y = 10 . . . (2)

From equation (1) we get

x = 5 - y

Now, Substitute (x = 5 - y) in equation (2),

**(Step 2)**3(5 - y) - 2y = 10

15 - 3y - 2y = 10

15 - 5y = 10

-5y = 10 - 15

-5y = -5

y = 5/5

y = 1

So, the value of y is 1.

Now, Substitute the value of y in equation 1 to find x.

**(Step 3)**x + y = 5

x + 1 = 5

x = 5 - 1

x = 4

Hence, x = 4, y = 1, is the required solution.

### 2. Method of Elimination

In this method, we eliminate one of the two variables to obtain an equation in one variable.

**Step 1:**Multiply both the equations by such a number so as to make the coefficients of one of the two equations numerically the same.

**Step 2:**Now add or subtract the two equations to get an equation containing only one (variable). You need to solve the equation to get the value of the unknown variable.

**Step 3:**Now the last step is, substitute the value of the variable in the original equations and by solving that equation you will get your answer.

**Example:**Solve 3x + 2y = 11, 2x + 3y = 4

**Solution:**The given equations are,

3x + 2y = 11 . . . (1)

2x + 3y = 4 . . . (2)

Multiply equation (1) by 3

9x + 6y = 33 . . . (3)

Multiply equation (2) by 2

4x + 6y = 8 . . . (4)

Now, subtract equation (4) from equation (3)

5x = 25

x = 25/5

x = 5

So, The value of x = 5

Now, put the value of (x = 5) in equation (2), we get

2x + 3y = 4

10 + 3y = 4

3y = -6

y = -6/3

y = -2

Hence, (x = 5) and (y = -2) is the required solution.

**Also Read:**

Linear equations in two variables questions