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# System Of Equations

A System of Equations is basically a collection of Linear Equations and includes the same Set of the variables in each and every equation of the system. This system of equations is also known as the Linear System. For instance:

2x + 6y + z = 0

4x + 7y – 2z = 2

7x – 4y + z = -1

This is a system of equation. Here in this example there are three equations with the same set of variables which are x, y, z and are used in all the equations. A trick to solve a linear system is that we should assign numbers to the variables such that it satisfies all the equations.

The linear system is a branch of the linear Algebra.

The general system of n linear equations with m unknowns can be given as:

A11 X1 + A12 X2 + . . . . . + A1m Xm = B1

A21 X1 + A22 X2 + . . . . . + A2m Xm = B2

. . . . . . . . . . . . . . . . . . . . . . . .. . . . . .  .

. . . .. . . . . . . . . . . .. . . . .. . . . . . . .. .  .

An1 X1 + An2 X2 + . . . . . + Anm  Xm = Bn

Here in above system X1, X2, . . . . . . ,Xm are the unknowns and A11, A12 . . . . . . , Anm are the coefficients of all the unknowns and B1, B2. . . . . , Bn are constants.

Although the unknowns and the coefficients are the complex or Real Numbers but they may also be integers or Rational Numbers.

Now here are the properties of the system of linear equations:

1.      Independence: In any system of the equations the equations must be independent from each other if none of the equations will be derived from the other equations. If the equations do not depend on each other than each will consist of new information about the variables which are used and if we remove any equation then it will simply increase the time and size of the solution.

2.      Consistency: This is also a very important property of the system of linear equations. The inconsistency in the equation occur if the left side terms of the equations are linearly depends on each other and the right hand side terms which are the constant terms will not satisfy the dependency relationship.

A system is consistent if the terms at left hand side are linearly independent form each other.

3.      Equivalence: In systems of equations; Two linear systems which are using the same set of variables in their equations if and only if each equation of one system can be derived from the equations of the other system and vice versa. Also if they have same solution set then also two systems can be said equivalent.

## Solving Systems by Substitutions

As we know there are several methods of solving Systems of Equations Like elimination method, substitution method and cross multiplication method. As compared to all these methods, Solving Systems by Substitution method is easier.
Before discussing the way to solve the System by Substitution method first let us try to understand the exact meaning of sub...Read More

## Solving Systems by Elimination using Multiplication

To solve any System of Equations, we Mean that we want to find the value of the unknown variables from the two equations. While using the method of elimination, we mean that one of the variables is first eliminated from the Set of the equations. For this we need to first check the co-efficient of any variable is same or not. Now if we ...Read More

## Solving Systems by Elimination using Addition and Subtractions

When we solve any equation, it means that we are going to find the value of the variable given in the equation.  This solution can be found by different ways.  In case we have two equations with two variables, in such equations, we need to find the value of 2 variables with the help of two equations. This is done using either ...Read More

## Solving Systems by Graphing

While Graphing a line we get a Straight Line then the line equation is known as linear equation or mathematically the equation of line is:
⇒ y = mx + c;
In the given equation, ‘m’ is the Slope of a line and ‘c’ is the y- intercept.
Now we will solve the system by graphing. For solving systems by graphing we have to follow some the steps so that we can easi...Read More

## Solving Systems of Inequalities by Graphing

If variables are not equals to each other then we can say it is an inequality.
Now we will see some of the conditions for an inequality which are:
Suppose we have two variables ‘s’ and ‘t’ then:
Condition1: If s ≠ t; which denotes ‘s’ is not equals to ‘t’;
Condition 2: If s < t then it represents ‘s’ is less than ‘t’;
Condition3: If s > t...Read More