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 :

Question : 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

Solution : The given linear equation is 2x + 3y – 8 = 0

(i) For intersecting lines, we know that

\(a_1\over a_2\) \(\ne\) \(b_1\over b_2\)

Any intersecting line may be taken as

5x + 2y – 9 = 0

(ii) For parallel lines, \(a_1\over a_2\) = \(b_1\over b_2\) \(\ne\) \(c_1\over c_2\)

\(\therefore\) line parallel to 2x + 3y – 8 = 0 may be taken as

6x + 9y + 7 = 0

(iii) For coincident lines, \(a_1\over a_2\) = \(b_1\over b_2\) = \(c_1\over c_2\)

Any line coincident to 2x + 3y – 8 = 0 may be taken as

4x + 3y – 16 = 0

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