The standard form of the system of linear equations in two variables x and y
a
_{1} x + b
_{1} y + c
_{1} = 0
a
_{2} x + b
_{2} y + c
_{2} = 0
Where a
_{1},b
_{1},c
_{1},a
_{2},b
_{2} and c
_{2} are all real numbers.
These pair of linear equations when plotted on a graph define two lines and only one of the below three possibilities can happen
a) Two lines intersect at one point
If
$\frac{a_{1}}{a_{2}}$ $\neq$
$\frac{b_{1}}{b_{2}}$, then the system of linear equations
a
_{1} x + b
_{1} y + c
_{1} = 0
a
_{2} x + b
_{2} y + c
_{2} = 0 has a unique solution that is exactly one solution and the equations are said to be a consistent pair of linear equations.
b) The two lines coincident with each other, that is they lie exactly on top of each other.
If
$\frac{a_{1}}{a_{2}}$ =
$\frac{b_{1}}{b_{2}}$ =
$\frac{c_{1}}{c_{2}}$, then the system of linear equations
a
_{1} x + b
_{1} y + c
_{1} = 0
a
_{2} x + b
_{2} y + c
_{2} = 0 has an infinite solution that is exactly one solution and the equations are said to be consistent pair of linear equations.
c) Two lines are parallel to each other, that is they will not intersect
If
$\frac{a_{1}}{a_{2}}$ =
$\frac{b_{1}}{b_{2}}$ $\neq$
$\frac{c_{1}}{c_{2}}$, then the system of linear equations
a
_{1} x + b
_{1} y + c
_{1} = 0
a
_{2} x + b
_{2} y + c
_{2} = 0 has no solution and the equations are said to be inconsistent pair of linear equations.
Examples on Graphical Method of Solution
The following examples help you understand how to solve simultaneous equations using the graphical method.
Example 1: Solve by graphing. The system given below has a unique solution (consistent system).
x  y = 2
x + y = 4
Solution :
For equation 1
x  y = 2
Subtract x on both sides
x  y  x = 2  x
y = 2  x
Multiply with (1) on both sides
(1) $\times$ (y) = (1)(2  x)
y = 2 + x
y = x  2
Now let us assign some values and find the corresponding yvalues to plot on the graph
x

0 
2

3 
y = x  2

y = x  2 = 0  2 = 2 
y = x  2
= 2  2
= 0

y = x  2
= 3  2
= 1 
For equation 2
x + y = 4
Subtract x on both sides
x + y  x = 4  x
y = 4  x
Let us assign some values and find the corresponding yvalues to plot on the graph
x 
0

2 
3

y = 4  x 
y = 4  x = 4  0 = 4

y = 4  x = 4  2 = 2 
y = 4  x = 4  3 = 1 
Now let us plot these ordered pairs
The two linear graphs intersect at the point (3, 1).
Example 2: Solve by graphing. The system given below has a no solution (inconsistent system).
2x + y = 5
2x + y = 8
Solution :
For equation 1
2x + y = 5
Subtract 2x on both sides
2x + y  2x = 5  2x
y = 5  2x
Now let us assign some values and find the corresponding yvalues to plot on the graph
x

0 
2

4

y = 5  2x

y = 5  2x = 5  2(0) = 5

y = 5  2x
= 5  2(2)
= 1 
y = 5  2x
= 5  2(4)
= 3 
For equation 2
2x + y = 8
Subtract 2x on both sides
2x + y  2x = 8  2x
y = 8  2x
Let us assign some values and find the corresponding yvalues to plot on the graph
x

0

2

4 
y = 8  2x

y = 8  2x = 8  2(0) = 8

y = 8  2x
= 8  2(2)
= 4 
y = 8  2x
= 8  2(4)
= 0 
Now let us plot these ordered pairs
From the above we can see that two lines are parallel to each other. So the system has no solution.