Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(ii) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
(i) Given, 2x + y – 6 = 0 and 4x – 2y – 4 = 0
(a1/a2) = 2/4 = 1/2
(b1/b2) = 1/-2
(c1/c2) = -6/-4 = 3/2
Since, (a1/a2) ≠ (b1/b2)
The given linear equations are intersecting each other at one point and have only one solution. Hence, the pair of linear equations is consistent.
Now, for 2x + y – 6 = 0 or y = 6 – 2x
x \(\rightarrow\) 0, 1, 2
y \(\rightarrow\) 0, 4, 2
And for 4x – 2y – 4 = 0 or y = (4x-4)/2
x \(\rightarrow\) 1, 2, 3
y \(\rightarrow\) 0, 2, 4
So, the equations are represented in graphs as follows:
From the graph, it can be seen that these lines are intersecting each other at only one point,(2,2).
(ii) Given, 2x – 2y – 2 = 0 and 4x – 4y – 5 = 0
(a1/a2) = 2/4 = 1/2
(b1/b2) = -2/-4 = 1/2
(c1/c2) = 2/5
Since, a1/a2 = b1/b2 ≠ c1/c2
Thus, these linear equations have parallel and have no possible solutions. Hence, the pair of linear equations are inconsistent.