**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.**