Solution of Linear System

Solution of linear system:

The linear system is the collection of two linear equations which have the same set of variables. By solving these two linear equation we can get solution of the linear system. We can use different method to find the solution to the linear system of equations, they are

Elimination method
Substitution method

Depending  up on the solutions , the linear systeems are classified into

Independent system with one solution point.

inconsistent system with no solution point (parallel line)

Dependent sytem

Methods to Find the Solution of the Linear System:
Elimination method:

Elimination method is similar to the addition method of solving the linear system of the equation.

Substitution method:

In this method we solve for one equation for one of the variable and then substitute the value obtained in the second equation

Model problems:

1.  Find the solution of the linear system of the equations by the Elimination method?

The equations are

2x+y= 3

x-y= 3

Solution:

Add both the equations

2x+y =3

x-y =3

3x=6

x=2

Here when we add the two equations, y terms get cancelled out and we get x=2

Now plug in x=2 in equation x-y=3

2-y=3

-y= 3-2

-y=1

y=-1

The solution of the linear system of equation (2,-1) (independent system with one solution point)

Model Problems Showing Solving for Solution of Linear System

Example2.

Find the solution of the linear system of the equations by substitution method?

The equations are

2x+y= 3

x-y= 3

Solution

First solve the one equation for one of the variable, that is

x-y = 3

-y=3-x

y=x-3

The value obtained is y=x-3

Plug in y=x+2 in the equation 2x+y=3

2x+x-3=3

3x-3=3

3x= 3+3

3x= 6

x=2

Now plug in x=2 in the equation y=x-3

y=2-3

y= -1

The solution of the linear system of the equation is (2,-1) (independent system with one soution point)

Example:3 Find the solution of the linear system of the equations by substitution method?

The equations are

2x+y= 6

x+y= 3

Solution

First solve the one equation for one of the variable, that is

x+y = 3

y=3-x

y=-x+3

The value obtained is y=-x+3

plug in y=-x+3 in the equation 2x+y=6

2x-x+3 = 6

x+3 =6

x=6-3

x=3

Now plug in x=3 in y=-x+3

y=-3+3

y=0

The solution of the linear system of the equation is (3, 0) (independent system with one soution point)