Why is linear equations a difficult topic to some students? The problem is that sometimes students did not get a clear picture on the basic concepts before they start solving linear equations questions. But still, it is a great experience for information and helps students to understand what linear equations are and where they should focus. We have already discussed about the various methods of solving linear equations. In mathematics, graphs are one the most useful tools to represent informations in a meaningful way. Let us discuss with graphing systems of linear equations in two variables along with step by step solutions of various problems. 

         

How to Graph Linear Equations

The graph of linear equations in two variables will be a straight line. Follow the below steps to plot the graph in a coordinate system:

Step 1: Write the given equation in the form: y = ax + b.

Step 2: Consider different value of x and substitute it in the given equation and get the corresponding values for y.

Step 3: Plot the points on a graph.

Step 4: Join all the points.

Solving Linear Equations by Graphing

In algebra of graphing linear equations we find the co-ordinates of the points on each line. The method of graphing the system of linear equation is also called "Graphing linear equations ".
Since the domain consists of all real numbers, we plug in values for x and find the corresponding values of y.  This will definitely help us to decide if the system of equations has unique solution, many solutions or no solution.

By graphing linear equations the solution to the linear equations with two variables will be as follows.
 
1. If the graph of the pair of linear equations are of intersecting lines then the given system of linear equations is consistent and have unique solution ( one and only one solution)

2. If the graph of the pair of linear equations are of parallel lines then the system of linear equations is inconsistent and have no solution.

3. If the graph of the pair of linear equations are of coinciding lines then the system of linear equations is consistent and have infinite number of solutions.
In Short,

If two are lines are Intersecting  Unique Solution  System is Consistent 
If two are lines are Parallel No Solution System is Inconsistent
If two are lines are Coinciding Infinite Solutions System is Consistent

Examples


Let us study the following system of equations and their respective graphs.
Example 1: Solve x - 2y = 1 ; x + y = 4
Solution: Find the x and y intercepts and record it in the table given below.
x - 2y = 1
   x  
   y   
   0
   -0.5 
   1       0

x + y = 4
Let us find the x and y intercepts and record it in the tables given below.
    x   
   y     
    0
   4
    4
   0
Solve Linear Equations using Graphing Method
     
By plotting the points we see that the system of equations intersect at ( 3, 1).
Hence the solution is (3,1)

Example 2: Solve 3x - 2y = 6,  6x - 4y = -12
Solution: Let us find the x and y intercepts and record it in the tables given below.
3x - 2y = 6
    x    
   y    
    0
   -3
    2
   0

6x - 4y = -12

     x   
     y    
    0
    3
    -2     0
 
Graphing Linear Equations Examples


From the graph we observe that the lines are parallel, Hence we say that the system of equations has "no solution".

Example 3: Solve 5x + 2y = 10 ;  10 x + 4 y = 20
Solution:
Assume the values of x and find the value of y. 5 x + 2y = 10
   x  
    y  
   0
   5
   2    0

10 x + 4y = 20

     x    
      y     
   -2     10
     4
    - 5

 Examples on Graphing Linear Equations


By plotting the above points and graphing we observe that the pair of lines coincide.
This shows that the system of given equation has infinite number of solutions.

Practice Questions

Solve the following system of linear equations graphically.
Problem 1. x + y = 3 ; 3x + 5y  = 15.

Problem  2. 3x - y = 2, 9x - 3y = 6

Problem 3. 4x - 5y =  4 and 8x - 10 y = 40