Systems of equations are also known as simultaneous equations. In algebra, systems of equations are the finite set of equations to be solved together. Solving systems of equations in real life involve age-related problems, modeling problems, etc where the unknown values can be represented in the form of variables.

## Classifications of System of Equations

The classification of the system of equations is similar to that of a single equation. Namely

- System of linear equations
- System of nonlinear equations:
- System of bilinear equations:
- System of polynomial equations,
- System of differential equations, or a
- System of difference equations

### Types of Solutions for the System of Linear Equations

System of linear equations is a set of linear equations involving the same set of variables. Ex: 3x – 4 = 5. The solution found for the system of linear equations should make all the equations true simultaneously.

The system of equations may or may not have a solution. If the solution exists for a system of equations then it is called a consistent system otherwise inconsistent system.

Ex: x+3y = – 3 and x+3y=3. The values of x and y cannot make both equations true simultaneously. Hence this system of equations is inconsistent.

#### Methods to Find the Solution for the System of Linear Equations

The solution found for the system of equations should satisfy all the conditions of the equations. Mainly there are three different methods to find the solution for the system of linear equations.

They are, Substitution Method, Elimination Method, and Graphical Method. Let us discuss each one of these methods in detail.

### 1. Substitution method:

Solve: 2x −3y = 6 and x + 3y = 48 using substitution method

2x −3y = 6 (1)

x + 3y = 48 x = 48 – 3y

Substitute the value of x in equation 1

2(48 – 3y) – 3y = 6

96 – 6y – 3y = 6

96 – 9y = 6 96 – 6 = 9y 90 = 9y

y = 10

Substitute the value of y in x equation

x = 48 – 3y

x = 48 – 3 (10) x = 48 – 30

x = 18

Thus in the substitution method, we find the solution of the system of equations by solving the equation for one variable and then substituting it in the other equation to find the values of the variables.

### 2. Elimination method:

Solve: 2x −3y = 6 and x + 3y = 48 using Elimination method

2x – 3y = 6

+(x + 3y = 48)

Here when you add these 2 equations you get,

3x = 54

x = 18

Now by substituting the value of x in any of the given equations we get the value of y.

2x – 3y = 6 2(18) – 3y = 6 36 – 3y = 6

36 – 6 = 3y 30 = 3y

y = 3

As the name indicates we eliminate a variable here to find the solution.

### 3. Graphical method:

Solve: 2x −3y = 6 and x + 3y = 48 using graphical method

Take the first equation 2x – 3y = 6. find two lowest values of x and y which satisfies the equation.

x = 3, y = 0 and x = 6, y = 2.

Similarly, find two lowest values of x and y which satisfies the second equation x + 3y = 48.

X = 45 , y = 1 and x = 42, y = 2.

Now you have two points P(3,0) and Q(6,2) for the first equation and R(45,1) and S(42,2) for the second equation. Plot the graph using these points. The point at which both the lines cross each other gives the solution for the given system of equations.

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