On comparing the ratios $straight a subscript 1 over straight a subscript 2 comma straight b subscript 1 over straight b subscript 2 space and space straight c subscript 1 over straight c subscript 2$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.

9x + 3y + 12 = 0, 18x + 6y + 24 = 0

Solution

straight a subscript 1 equals 9 comma space straight b subscript 1 equals 3 comma space straight c subscript 1 equals 12 straight a subscript 2 equals 18 comma space straight b subscript 2 equals 6 comma space straight c subscript 2 equals 24 therefore straight a to the power of 1 over straight a subscript 2 equals 9 over 18 equals 1 half comma space straight b subscript 1 over straight b subscript 2 equals 3 over 6 equals 1 half rightwards double arrow space straight a subscript 1 over straight a subscript 2 equals straight b subscript 1 over straight b subscript 2 equals straight c subscript 1 over straight c subscript 2

Thus, the lines representing the pair of linear equations coincide.

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