Solving Systems of Linear Equations

A linear equation is a first degree equation having one or more  variables. A system of linear equations is formed by two or more linear equations with the same variables. Solution of a system of linear equations in two variables is an ordered pair of values of the variables that satisfy each of the equation in the system. The equations of a system are often called simultaneous equations as any solution to the system has to satisfy all the equations simultaneously. The equations cannot be solved independently of one another.

Solving Systems of Linear Equations Solver

The systems of linear equations are categorized into two types on the basis of their solutions.
1. Consistent system
A consistent system is subdivided into two types
• System with unique solution
• System with many solutions
2.   Inconsistent system
• Inconsistent systems have no solution.

Methods of Solving a System of Linear Equations

The important methods applied in solving a system of linear equations are
• Graphical Method
• Substitution Method
• Elimination Method
• Cramer's Rule
• Matrix Row ellimination

Graphical Method of Solving Linear Equations

A graphical solution is obtainable for two  linear equations by plotting them on cartesian coordinates with one axis corresponding one variable and the other to another variable. In graphical methods graphs representing the equations in cartesian system. The point of intersection of all the lines drawn gives the solution to the system.
• If the lines intersect at a point, then the system is consistent and has a unique solution.
• If the lines drawn parallel, there will not be a point of intersection. The system is inconsistent and no solution exists for the system.
• If the lines are coincident, the system is consistent and have many solutions.
Solved Example
Question: Solve graphically

2x + y = 5

4x - y = 1

Solution:
System of linear equations is given as

2x + y = 5

4x - y = 1

The Graphs of the equations to be straight lines. The solution set to the system is the set of all ordered pairs that satisfy both equations. If we graph each equation on the same set of axes, then we can see the solution set.

Graphical representation of given system:

Point of intersection of two lines is the solution of linear system that is (1, 3).

Solving System of Linear Equations By Substitution

Substitution method used for solving a system of linear equations. In substitution method, one of the variables is solved literal in terms of the other variable. This in turn is substituted in the second equation to reduce it to an equation in one variable.
Solved Example
Question: Solve

y = 6x -11

-2x - 3y = -7

Solution:
Given linear system

y = 6x -11     .........(1)

-2x - 3y = -7    .........(2)

Here the first equation is already solved for y. The expression for y is substituted in the second equation.

-2x - 3(6x - 11) = -7

=> -2x - 18x + 33 = -7

=> -20x + 33 = -7

=> -20x + 33 - 33 = -7 - 33

=> -20x = -40

=> x = 2

Substituting x =2 in equation (1)

y = 6(2) - 11

= 12 - 11

=1

=> y = 1.

Solving System of Linear Equations By Elimination

In elimination method one variable is eliminated manipulating the equations and adding (or subtracting) them. Since the addition removes one variable, this method is also known as addition method.
Solved Example
Question: Solve x + y = 13

x - y = 3

Solution:
Given
x + y = 13    .........(1)

x - y = 3       .........(2)

Here the variable y can be eliminated by adding the two equations.

(1) + (2)

=> 2x = 16

=> x = 8

substituting x = 8 in equation (1)

x + y = 13

=> 8 + y = 13

=> y = 13 - 8

=> y = 5

Cramer's  Rule

Cramer's rule is a method, based on determinants, for solving systems of simultaneous linear equations. The given system is considered as a matrix arrangement. Cramer's rule makes use of determinant function in soling the equation.  It is very useful in determining whether a system has a unique solution without actually solving it.

Cramer's rule for systems of two linear equations:

$a_1$x + $b_1$y = $c_1$

$a_2$x + $b_2$y = $c_2$

then the solution is given by

x = $\frac{D_1}{D}$  and

y = $\frac{D_2}{D}$

Where

D = $\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}$  $\neq$ 0

$D_1$ = $\begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix}$

$D_2$ = $\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix}$.
Solved Example
Question: Solve linear equations system by using Cramer's rule.

x + y = 3

2x - y = 1

Solution:
Given linear equations system

x + y = 3

2x - y = 1

D = $\begin{vmatrix} 1 & 1\\ 2 & -1 \end{vmatrix}$

= -1 - 2 = - 3 $\neq$ 0

$D_1$ = $\begin{vmatrix} 3& 1\\ 1 & -1 \end{vmatrix}$

= - 3 - 1

= - 4

$D_2$ = $\begin{vmatrix} 1& 3\\ 2 & 1 \end{vmatrix}$

= 1 - 6

= - 5

Now by Cramer's Rule:

The solution of given system is

x = $\frac{D_1}{D}$  and

y = $\frac{D_2}{D}$

=> x = $\frac{-4}{-3}$

= $\frac{4}{3}$

and y = $\frac{-5}{-3}$

= $\frac{5}{3}$