6. 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

Solutions: (i) Given the linear equation 2x + 3y – 8 = 0. To find another linear equation in two variables such that the geometrical representation of the pair so formed is intersecting lines, it should satisfy below condition; (a1/a2) ≠ (b1/b2) Thus, another equation could be 2x – 7y + 9 = 0, such that; (a1/a2) = 2/2 = 1 and (b1/b2) = 3/-7 Clearly, you can see another equation satisfies the condition. (ii) Given the linear equation 2x + 3y – 8 = 0. To find another linear equation in two variables such that the geometrical representation of the pair so formed is parallel lines, it should satisfy below condition; (a1/a2) = (b1/b2) ≠ (c1/c2) Thus, another equation could be 6x + 9y + 9 = 0, such that; (a1/a2) = 2/6 = 1/3 (b1/b2) = 3/9= 1/3 (c1/c2) = -8/9 Clearly, you can see another equation satisfies the condition. (iii) Given the linear equation 2x + 3y – 8 = 0. To find another linear equation in two variables such that the geometrical representation of the pair so formed is coincident lines, it should satisfy below condition; (a1/a2) = (b1/b2) = (c1/c2) Thus, another equation could be 4x + 6y – 16 = 0, such that; (a1/a2) = 2/4 = 1/2 ,(b1/b2) = 3/6 = 1/2, (c1/c2) = -8/-16 = 1/2 Clearly, you can see another equation satisfies the condition.