Given the linear equation \(2x + 3y – 8 = 0\), write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) Intersecting lines
(ii)Parallel lines
(iii)Coincident lines

Ans.(i) Intersecting lines

The condition for intersecting lines is :
\({{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}\)

So the second equation can be \(x + 2y = 3\)

Since, \({{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}\)
=>\({{2}\over {1}} \ne {{3}\over {2}}\)

(ii) Parallel lines

The condition for parallel lines is :

\({{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}\)

So the second equation can be \(2x + 3y – 2 = 0 \)

Since, \({{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}\)
=>\({{2}\over {2}} = {{3}\over {3}} \ne {{-8}\over {-2}}\)

(ii) Coincident lines

The condition for coincident lines is :

\({{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}\)

So the second equation can be \(4x + 6y – 16 = 0 \)
Since, \({{a_1}\over {a_2}} = {{b_1}\over {b_2}} = {{c_1}\over {c_2}}\)
=>\({{2}\over {4}} = {{3}\over {6}} = {{-8}\over {-16}}\)
=>\({{1}\over {2}} = {{1}\over {2}} = {{1}\over {2}}\)