A system of equations is a set of equations that we deal with all together at once. Linear equations are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables. If these two linear equations intersect, that point of intersection is called the solution to the system of linear equations.

The solution of system of linear equations in two variables is the point of intersection (x, y) of the graphs of the line.
The algebraic methods of solving a system of linear equations, the substitution and elimination methods can lead to an exact solution of the system. This against the graphical method of solving the system often gives approximate solutions as it is difficult to read the coordinates accurately. But still, the graphical method provides a visual proof to the existence of the solution.

Solution of a System of Linear Equations

Solution of a System of linear equations based upon the slope of the lines

The system has exactly one solution
Systems have 1 and only 1 solution when the two lines have different slope. If the two lines have different slopes then eventually at some point they must meet or lines are intersecting at any point.

System has no solutions
The Systems have no solution when the lines are parallel that is lines have the same slope and the lines have different y-intercepts.

The system has infinite solutions
The Systems have infinite solutions when the lines are parallel and the lines have the same y-intercept. If two lines are parallel and the same y-intercept, they are actually the same exact line.

Graphical Method of Solving Linear Equations

Two linear equations with two variables, considered at the same time is a system of linear equations and, of course, have graphs that are straight lines. The intersection point of the graphs to be a point whose coordinates are solutions to the system. The graph of a linear equation in two variables is a straight line in coordinate plane. Every point on the line is a solution of the equation.

Steps for Solving a System of Linear Equations Graphically:

Step 1: Write both the equations in slope intercept form.

Step 2: Make a table of values to be plotted for both the equations.

Step 3: Draw the graphs of the two equations.

Step 4:
Mark the point of intersection and read the coordinates (x, y).

Step 5: The Ordered pair (x, y) so obtained is the solution of the equation.

Solved Examples

Question 1: Solve the system of equations
$y + x = 3$ and $3y - x = 5$
Solution:
Step 1:
Write both the equations in slope intercept form isolating y on one side of the equation

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

Step 2:
Make a table of values to be plotted for both the equations

$y = - x + 3$
                                          
     x    
     y     
    1     2
   0     3
  -1     4


$y$ = $\frac{x + 5}{3}$


      x    
     y     
     1
    2
    -2     1
    4    3

Even though two points are sufficient to draw a line, three points are taken to ensure there is no error done in graphing.

Step 3:
Draw the graphs of the two equations





The graph of the lines are shown with the points marked. The green line represents the equation y = - x + 3 and the purple line is the graph of 3y - x = 5.

Step 4:
Mark the point of intersection and read the coordinates (x, y)

=> Lines intersect at the point (1, 2)

Hence the solution of the system is (1, 2) where x =1 and y = 2.

 

Question 2: Solve graphically the system

$y - x = 3$  and  $y + x = -1$
Solution:
Step 1:
Find the x and y intercepts of the given lines

y = 3 + x  and  y = -1 - x

Step 2:
Make a table of values to be plotted for both the equations

y = 3 + x

   x        y  
   0
   3
  -3
   0
  -2
   1

and y = -1 - x

   x    
     y   
   0
    -1
  -1
     0
   -2
     1


Step 3:
Draw the graphs of the two equations





The intercepts are plotted and the graph of the lines made joining the respective intercepts.The green line is the graph of y - x = 3 and the the purple line represents y + x = -1.

Step 4:
The above graph shows the point of intersection as (-2,1)

Hence the solution to the system is x = -2 and y = 1.