As shown in the above example, we compute the variable value from one equation and substitute it into the other.
We are given that
y = 24 – 4x ------(1)
2x + y/2 = 12 ------(2)
Here we choose equation (1) to compute the value of x. Since equation (1) is already in its
most simplified form:
(Putting this value of y into equation (2) and then solving for x
2x + (24-4x)/2 = 12 ------(2) (∵ y = 24 – 4x)
2x + 24/2- 4x/2 = 12
2x + 12 – 2x = 12
12 = 12
You might feel that this is the same scenario as discussed above (that of 24 = 24). But
wait! You are trying to jump at a conclusion a bit too early. In the previous scenario, the
result 24 = 24 had resulted because we put the variable value into the same equation that we
used for its computation. Here we have not done that.
The result 12 = 12 has got something to do with the nature of the system of equations that we
are given. No matter what solving technique you might be using, a solution to a system of linear
equations lies at a single point where their lines intersect. In this scenario, the two lines
are basically the same (one line over the other. The following figure shows this scenario.
Such a system is called a dependent system of
equations. And solution to such a system is the entire line (every point on the line is a point
of intersection of the two lines)
Hence the solution to the given system of equations is the entire
line: y = 24 – 4x
Another possible Scenario:
Similar to this example, there exists another scenario where substitution of one variable
into the 2nd equation leads to a result similar to one shown below:
23 = –46
5 = 34
Such a scenario arises when there exists no solution to the given system of equations. I.e.,
when the two lines do not intersect at any point at all.
Hence in case of such a result, where your basic Math rules seem to fail, a simple conclusion
is that no solution to the given system exists. Such a system of equations is called an Inconsistent system.