Let the system of pair of linear equations be

a_{1}x + b_{1}y = c_{1} ....(1)

a_{2}x + b_{2}y = c_{2} ....(2)

We know that given two lines in a plane, only one of the following three possibilities can happen -

- The two lines will intersect at one point.
- The two lines are coincident lines.
- The two lines will not intersect, however far they are extended, i.e., they are parallel.

Example: The path of highway number 1 is given by the equation x + y = 7 and the highway number 2 is given by the equation 5x + 2y = 20. Represent these equations geometrically.

**Solution:** We have, x + y = 7

⇒ y = 7 – x ....(1)

In tabular form

Points | A | B |

x | 1 | 4 |

y | 6 | 3 |

and 5x + 2y = 20

⇒ y = (20 - 5x)/2 ....(2)

In tabular form

Points | C | D |

x | 2 | 4 |

y | 5 | 0 |

Plot the points A (1, 6), B(4, 3) and join them to form a line AB.

Similarly, plot the points C(2, 5). D (4, 0) and join them to get a line CD. Clearly, the two lines intersect at the point C. Now, every point on the line AB gives us a solution of equation (1). Every point on CD gives us a solution of equation (2).

Example: A father tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Represent this situation algebraically and graphically.

Solution: Let the present age of father be x-years and that of daughter = y years

Seven years ago father’s age

= (x – 7) years

Seven years ago daughter’s age

= (y – 7) years

According to the problem

(x – 7) = 7(y – 7)

or x – 7y = – 42 ....(1)

After 3 years father’s age = (x + 3) years

After 3 years daughter’s age = (y + 3) years

According to the condition given in the question

x + 3 = 3(y + 3)

or x – 3y = 6 ....(2)

x | 0 | 7 | 14 |

y = (x + 42)/7 | 6 | 7 | 8 |

Points | A | B | C |

x – 7y = –42

x – 3y = 6

x | 6 | 12 | 18 |

y = (x − 6)/3 | 0 | 2 | 4 |

Points | D | E | F |

Plot the points A(0, 6), B(7, 7), C(14, 8) and join them to get a straight line ABC. Similarly plot the points D(6, 0), E(12, 2) and F(18, 4) and join them to get a straight line DEF.

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