## Introduction:

We are already familiar with solving linear equations of one variable. Let us discuss the situation as follows. 4 men and 4 boys can do a piece of work in 3 days while 2 men and 5 boys can finish it in 4 days. How long would it take 1 boy to do it? How long would it take 1 man to do it? .

The above situation seems to be hard to solve. But using the method of solving simultaneous equations of two variables we can solve it.

In this topic let us discuss with the linear equations with 2 variables, types of solutions and the methods of solving the linear equations in 2 variables.

a

_{1}x + b

_{1}y = c

_{1}and

a

_{2}x + b

_{2}y = c

_{2}

The above system of linear equations of two variables satisfy any one of the following conditions.

1. One solution.

2. No solution

3. Infinitely many solutions

The above system of equations are also called simultaneous linear equations in two variables. If we look at the methods of solving linear equations in two variables, any one of the following methods can be used.

1. Substitution Method.

2. Elimination Method.

3. Cross multiplication Method

4.. Graphical Method

5. Matrix Method

6. Method of Determinants (Cramer's Rule)

and many more methods

### How to do linear Equations in two variables:

The linear equations in two variables can be solved by any one of the above methods. We should make sure that we frame the equations according to the standard form, ax + by + c = 0

While solving system of linear equations in two variables, we need to remember the following points.

The graph of the linear equations is a straight line.

When we graph the lines on a graph, one of the following conditions will be satisfied.

1. If the lines intersect at a point there will be one and only one solution. The solution is written in the form (x,y).

2. If the lines are parallel there will be no solution, as the lines do not meet.

3. If the lines coincide there will be infinite number of solutions.

**Consistency**: We say that the system of equations is consistent if it has at least one solution.

Intersecting lines are coinciding lines are the consistent system equations.

**Inconsistency**: We say that the system of equations is inconsistent if it has no solution.

Parallel lines are inconsistent system of equations.

### Linear Equations in Two Variables:

When we consider the linear equations in two variables, their solutions will satisfy any one of the following conditions.

1. Only one solution (unique solutions)

2. No solution

3. Infinitely many solutions satisfying the given equation.

we shall discuss the more examples of linear equations in two variables in this section.

### Linear Inequalities in two variables:

When we consider the linear inequalities in two variables, the expressions and the constants are related by the sign >, <, <= or > =.

Let us consider the following inequalities.

x+ y <= 4 and 2x + 3y > = 9

The region common to both the shaded region (checked in black and red ) is the solution to the above inequality.