(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

**Solution:**

**(i) Given, x + y = 5 and 2x + 2y = 10**

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

(a_{1}/a_{2}) = 1/2

(b_{1}/b_{2}) = 1/2

(c_{1}/c_{2}) = 1/2

Here, (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

∴The equations are coincident and they have an infinite number of possible solutions.

So, the equations are consistent.

For, x + y = 5 or y = 5 – x

x | 0 | 3 | 5 |

y | 5 | 2 | 0 |

For 2x + 2y = 10 or y = (10 – 2x)/2

x | 0 | 3 | 5 |

y | 5 | 2 | 0 |

So, the equations are represented in graphs as follows:

From the figure, we can see, that the lines are overlapping each other.

Therefore, the equations have infinite possible solutions.

**(ii) Given, x – y = 8 and 3x – 3y = 16 **

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

(a_{1}/a_{2}) = 1/3

(b_{1}/b_{2}) = -1/-3 = 1/3

(c_{1}/c_{2}) = 8/16 = 1/2

Here, (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2})

The equations are parallel to each other and have no solutions. Hence, the pair of linear equations is inconsistent.

**(iii) Given, 2x + y – 6 = 0 and 4x – 2y – 4 = 0 **

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

(a_{1}/a_{2}) = 2/4 = ½

(b_{1}/b_{2}) = 1/-2

(c_{1}/c_{2}) = -6/-4 = 3/2

Here, (a_{1}/a_{2}) ≠ (b_{1}/b_{2})

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 | 0 | 1 | 2 |

y | 6 | 4 | 2 |

And for 4x – 2y – 4 = 0 or y = (4x – 4)/2

x | 1 | 2 | 3 |

y | 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).

**(iv) Given, 2x – 2y – 2 = 0 and 4x – 4y – 5 = 0**

_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

(a_{1}/a_{2}) = 2/4 = ½

(b_{1}/b_{2}) = -2/-4 = 1/2

(c_{1}/c_{2}) = 2/5

Here, (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2})

Thus, these linear equations have parallel and have no possible solutions. Hence, the pair of linear equations are inconsistent.