Solving Linear Equation using Substitution Method

Substitution Method is another method of solving any linear equation. To understand much better we shall go through an example and also graphical representation showing how to solve linear equation using substitution method.

Let us Consider the following system of linear equations:
x - y = -2
x - y = 1.

Using the method of substitution, we subtract the second equation from the first to obtain: 0 = -3. This is a false statement and the system, therefore, has no solution. If we look closer at the lines we see that they satisfy the condition

a₁/a₂ = b₁/b₂

and are therefore parallel(as can be shown below). They do not intersect explaining why the system of linear equations has no solution.














What if the two equations represent the same line?

Consider the equations

x - y = 1
2x - 2y = 2

Multiply the first equation by 2 to put the equations in the form
2x - 2y = 2
2x - 2y = 2

Now subtraction gives 0 = 0, which is true no matter what values x and y may have! This time the two equations represent the same line, since both can be written in the form y = x - 1.














Any point on this line has coordinates which will satisfy both equations, so there are an infinite number of solutions!

In general, two equations represent the same line if one equation is a multiple of the other. That is a₁/a₂ = b₁/b₂ = c₁/c₂

There are then three possibilities for a pair of simultaneous linear equations:

(i) Just one solution (the usual situation - both lines are unique and not parallel to each other)
(ii) No solution ( the lines are parallel a₁/a₂ = b₁/b₂)
(iii) Infinitely many solutions (the equations represent the same line, a₁/a₂ = b₁/b₂ = c₁/c₂