A set of equations for which we get a common solution is known as system of equations. System of equations can be either linear or non-linear.

Two set of equations will have two variables, three set of equations will have three variables and so on. System of equations can have any number of variables which can be linear or non-linear.

A system of equations is a collection of two or more unknowns where we try to find values for each of the unknowns which satisfies every equation in the system. Linear System of Equations

Linear equations use only linear functions and operations. Exponents will not be higher than one. When graphed, it will be a straight line.
Example: x + y + z = 6
They are mostly used in analysis.

Non-Linear System of Equations

It can consist of trignometric functions, multiplication or division by variables. One or more of the variables may contain an exponent larger than one. When graphed it represents some sort of curves.
Example: $x ^{2}$ + $y^{2}$ + z = 4
Non-linear equations dominate the realm of higher math and science.

A general system of m linear equations with n unknowns can be written as

$a_{11}x_{1}$ + $a_{12}x_{2}$ + .........+ $a_{1n}x_{n}$ = $b_{1}$
$a_{21}x_{1}$ + $a_{22}x_{2}$ + .........+ $a_{2n}x_{n}$ = $b_{2}$
...... ........
....... .......
$a_{m1}x_{1}$ + $a_{m2}x_{2}$ + .........+ $a_{mn}x_{n}$ = $b_{m}$
There are three types of system of linear equations. They are:
  • Independence
  • Consistency and
  • Homogeneous

Independence: A system of linear equations are independent if there is no possibility of deriving equation from the others. When the equations are independent each equation will contain new information and removing any equation will lead to change in the size and solution set.

Consistency: A system of linear equations are consistent if they have a common solution else it will be inconsistent.
Example: -2x + 3y = 8
3x - y = -5
Given above is an example for consistent system of linear equations.

Homogeneous
: A system of linear equations are homogeneous if they can be derived algebraically with the same set of variables from the other else they are non-homogeneous.
Example: $x_{1}$ +$x_{2}$ - $2x_{3}$ = 0
$3x_{1}$ +$2x_{2}$ + $4x_{3}$ = 0
$4x_{1}$ +$3x_{2}$ + $3x_{3}$ = 0
System of equations can contain any number of variables and equations. There are three main methods to solve them, and you can use any one of them.
  • Substitution method.
  • Graphical method.
  • Elimination method.
There are two necessary conditions in solving system of equations by substitution

1. Number of equations should be equal to number of variables. If there are two variables, there must be two equations; 3 Variables = 3 Equations etc.,
2. One of the equations can easily be solved for one variable.

Given below are the steps to be considered for solving system of equations using substitution method.

1. Select one equation and isolate one variable and name it as first equation.
2. In second equation substitute the value of the first equation and solve for the variable.
3. Again by substituting in the given equations the value of the second variable is found. This process is continued until the values for the given variables are known.

Solved Example

Question: x + y =3
y - 2x = 5
Solution:
 
The given equations are :  x + y = 3  .................(1)
                                                    y - 2x = 5        ................(2)


Step -1 : From the first equation we isolate x (y can also be isolated).

$\Rightarrow$  x = 3 - y.

Step -2 : Substitute x = 3 - y in the second equation.

$\Rightarrow$   y - 2(3 - y) = 5

Step -3 : Solve for y 
                       y - 6 + 2y = 5

$\Rightarrow$ 3y = 11

$\Rightarrow$  y = $\frac{11}{3}$

Step -4 : As the value of y is known. The value of x can be easily found now by substituting in any of the given equations.
Substituting  in x = 3 - y we get y as

                  x = 3 -  $\frac{11}{3}$

                 3x = 9 - 11

                  x = $\frac{-2}{3}$

Therefore, x = $\frac{-2}{3}$ and y = $\frac{11}{3}$.

Verification can be done by plugging the values of x and y in one of the given equations.
 

When there are two equations graphical method is one of the easiest and most convenient method to solve system of equations. For complex numbers it is not reliable and is not the preferred method. In graphical method there can be one, none or infinitely many solutions. If for the given system of equations, we can graph a straight line then it possible to solve them graphically.

Graph the two lines and look for the point where they intersect(cross). The intersection point is termed as the solution.

Solved Example

Question: Using graphical method solve the system of equations:

x + 2y  = 3
4x + 5y = 6
Solution:
 
In order to graph them we solve each equation for y.

Consider the first equation, x + 2y  = 3 

Solve for y                
$\Rightarrow$   2y = 3 - x           

 y = $\frac{3}{2}$ -$\frac{x}{2}$

The second equation is
4x + 5y = 6
$\Rightarrow$ 5y = 6 - 4x

$\Rightarrow$  y = $\frac{6}{5}$ - $\frac{4}{5}$ x

Now plot  y = $\frac{6}{5}$ -$\frac{4}{5}$ x   and  y = $\frac{3}{2}$ -$\frac{x}{2}$

We get
Equations by Graphing
From the above graph we see that the two lines intersect at the point (-1, 2).
 

The elimination method of solving systems of equations is also called the addition method. To solve a system of equations by elimination we transform the system such that one variable "cancels out".

Solved Example

Question: Solve the system of equations using elimination method

6x - 5y = 10
x  + 5y = 12
Solution:
 
For the given system of equations we can easily cancel out the y terms by adding two equations together.

Add the two given equations:

6x - 5y = 10
x  + 5y = 12
_________________
7x + 0   = 22

From  the equation 7x = 22 we can now find x.

Therefore x= $\frac{22}{7}$

To find the value of y substitute x =$\frac{22}{7}$ in one of the given equations.

Substituting x = $\frac{22}{7}$ in x + 5y = 12, we get

$\frac{22}{7}$ + 5y = 12

Simplify and find the value of y

5y = 12 - $\frac{22}{7}$

$\Rightarrow$ 35y = 84 -22

$\Rightarrow$ y = $\frac{62}{35}$

Therefore, The solution is (x, y) =  ( $\frac{22}{7}$ , $\frac{62}{35}$ )