Hey learners, here we are going to start with another lesson "Linear equations in two variables". In previous lessons we have already learnt basic linear equations and linear equations in one variable.

In this lesson, you will learn what are linear equations in two variables with some examples? What are the methods to solve the linear equations in two variables and we will also solve some basic problems.

## 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,
1. Substitution method
2. Elimination method
3. 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.