Which of the following pairs of linear equations are consistent / inconsistent? If consistent, obtain the solution graphically:
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 16
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Answer
(i) x + y = 5; 2x + 2y = 10
a1/a2 = 1/2
b1/b2 = 1/2 and
c1/c2 = 5/10 = 1/2
Hence, a1/a2 = b1/b2 = c1/c2
Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.
x + y = 5
x = 5 - y
| x | 4 | 3 | 2 |
| y | 1 | 2 | 3 |
And, 2x + 2y = 10
x = 10-2y/2
| x | 4 | 3 | 2 |
| y | 1 | 2 | 3 |
Graphical representation
From the figure, it can be observed that these lines are overlapping each other. Therefore, infinite solutions are possible for the given pair of equations.
(ii) x – y = 8, 3x – 3y = 16
a1/a2 = 1/3
b1/b2 = -1/-3 = 1/3 and
c1/c2 = 8/16 = 1/2
Hence, a1/a2 = b1/b2 ≠ c1/c2
Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
a1/a2 = 2/4 = 1/2
b1/b2 = -1/2 and
c1/c2 = -6/-4 = 3/2
Hence, a1/a2 ≠ b1/b2
Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.
2x + y - 6 = 0
y = 6 - 2x
| x | 0 | 1 | 2 |
| y | 6 | 4 | 2 |
And, 4x - 2y -4 = 0
y = 4x - 4/2
| x | 1 | 2 | 3 |
| y | 0 | 2 | 4 |
Graphical representation
From the figure, it can be observed that these lines are intersecting each other at the only one point i.e., (2,2) which is the solution for the given pair of equations.
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
a1/a2 = 2/4 = 1/2
b1/b2 = -2/-4 = 1/2 and
c1/c2 = 2/5
Hence, a1/a2 = b1/b2 ≠ c1/c2
Therefore, these linear equations are parallel to each other and thus, have no possible solution. Hence, the pair of linear equations is inconsistent.
