Question : Which of the following pairs of linear equations are consistent obtain the solution in such cases graphically.
(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 10
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Solution :
(i) We have the given equations,
x + y = 5 \(\implies\) y = 5 – x
If x = 0, y = 5
If x = 5, y = 0
| x | 0 | 5 |
| y = 5 – x | 5 | 0 |
| Points | A | B |
and 2x + 2y = 10 \(\implies\) y = \(10 – 2x\over 2\)
If x = 0, y = 5
If x = 2, y = 3
If x = 5, y = 0
| x | 0 | 2 | 5 |
| y = \(10 – 2x\over 2\) | 5 | 3 | 0 |
| Points | C | D | E |
Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C, D and E points we get the line CD.
We see that both the lines in the graph are coincident. Therefore both equations have infinitely many solutions.
Hence, the pair of linear equations is consistent.
(ii) We have the given equations,
x – y = 8 \(\implies\) y = x – 8
If x = 0, y = -8
If x = 8, y = 0
| x | 0 | 8 |
| y = x – 8 | -8 | 0 |
| Points | A | B |
and 3x – 3y = 16 \(\implies\) y = \(3x – 16\over 3\)
If x = 0, y = \(-16\over 3\)
If x = 2, y = \(-10\over 3\)
| x | 0 | 2 |
| y = \(3x – 16\over 3\) | \(-16\over 3\) | \(-10\over 3\) |
| Points | C | D |
Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C and D points we get the line CD.
We see that both the lines are parallel. Therefore both equations have has no solution.
Hence, the pair of linear equations is inconsistent.
(iii) We have the given equations,
2x + y – 6 = 0 \(\implies\) y = 6 – 2x
If x = 0, y = 6
If x = 3, y = 0
| x | 0 | 3 |
| y | 6 | 0 |
| Points | A | B |
and 4x – 2y – 4 = 0 \(\implies\) y = 2x – 2
If x = 0, y = 1
If x = -2, y = 0
| x | 0 | 1 |
| y | -2 | 0 |
| Point | D | E |
Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the straight line AB and by plotting C, D points we get the line CD.
The lines AB and CD intersect at E.
We see that both the lines in the graph have a common point E. Therefore the given equations is consistent and this point E gives us the solution of both the lines.
Hence, the pair of linear equations is consistent.
(iv) We have the given equations,
2x – 2y – 2 = 0 \(\implies\) y = x – 1
If x = 0, y = -1
If x = 1, y = 0
| x | 0 | 1 |
| y | -1 | 0 |
| Points | A | B |
and 4x – 4y – 5 = 0 \(\implies\) y = \(x – {5\over 4}\)
If x = 0, y = \(-5\over 4\)
If x = \(5\over 4\), y = 0
| x | 0 | \(-5\over 4\) |
| y | \(5\over 4\) | 0 |
| Points | C | D |
Now, plot these points in the table on graph as shown in figure below. By plotting A and B points we get the line AB and by plotting C and D points we get the line CD.
We see that both the lines are parallel. Therefore both equations have has no solution.
Hence, the pair of linear equations is inconsistent.
