Graphical Method of Solution of a Pair of Linear Equations
We know that there are infinitely many solutions of a linear equation in two variables. Each pair of values of x and y satisfying the equations is a solution and this pair represents a point on the graph of the equation.
Suppose we have two simultaneous equations in x and y, we can draw the graph for each of the equations. If the lines l_{1} and l_{2} represent these equations, the following possibilities arise.
(i) If l_{1} and l_{2} intersect at a point P (x_{1}, y_{1}), we say that the equations have a unique solution x = x_{1} and y = y_{1}.
(ii) If l_{1} and l_{2} are parallel then there is no common point between the lines and the equations have no solution.
(iii) If the lines are coincident then each point of one line, is also a point of the other line. In this case the equations have infinitely many solutions.
x + 2y = 5
3x  y = 1.
(i) x + 2y = 5
3x  y = 1.
x + 2y = 5.
â‡’ y =
âˆ´ If x = 1, y = = 3.
If x = 1, y = = 2.
If x = 3, y= = 1.
On the basis of the above, the following points.
x  1  1  3 
y  3  2  1 
Plotting (1, 3), (1, 2), (3, 1) and joining them, we get a straight line.
Now, 3x  y = 1
â‡’ y = 3x 1
If x = 1,
y = 3 1 = 4
If x = 1,
y= 3  1 = 2
If x = 2,
y = 6 1 = 5
We now have the following points
x  1  1  2 
y  4  2  5 
Plotting (1,4), (1, 2), (2, 5) and joining them, we get another straight line.
These lines intersect at the point P (1, 2) and therefore, the solution of the equation is x = 1, y =
Show graphically that these system of equations has no solution
x  2y = 4
2x  4y = 5
x  2y = 4
â‡’ y=
âˆ´ If x = 0, y = 2
If x = 2, y = 1
If x = 4, y= 0
We now get the following points.
x  0  2  4 
y  2  1  0 
Plotting (0, 2), (2, 1) (4, 0) and joining them, we get the required line.
2x  4y = 5
â‡’ y =
âˆ´ If x =
If , y = 1
If x = 2, y =
We now have the following points.
x   

2 
y  1 
Plotting, and joining them, we get the required line.
We observe that the given equations represent two parallel lines. As this has no common point between the lines, we can say that the above equations have no solution.
x + 2y = 4
3x + 6y = 12
x + 2y = 4
â‡’ y=
âˆ´ If x = 0, y = 2
If x = 2, y = 1
If x = 4, y = 0
We now have the following points.
x  0  2  4 
y  2  1  0 
Plotting A (0, 2), B (2, 1), C (4, 0) and joining them, we get the required line.
Now, 3x + 6y = 12
â‡’ y =
If x = 2, y =3
If x = 4, y = 4
If x = 6, y = 5
Plotting D (2,3), E (4,4), F (6,5) and joining them, we get the required lines.
We observe that all the points of
(i) (A, B, C) and
(ii) (D, E, F) lie on the same line.
Therefore, the line represented by (i) and (ii) coincide.
Consequently, every solution of (i) is a solution of (ii).
Hence the system of equations has infinite number of solutions.