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

6x - 3y + 10 = 0, 2x - y + 9 = 0.

Solution

Comparing the given equations with standard forms of equations a₁x + b₁y + c₁, = 0 and a₂x + b₂y + c₂ = 0 we have,

straight a subscript 1 equals 6 comma space straight b subscript 1 equals 3 comma space straight c subscript 1 equals 10 straight a subscript 2 equals 2 comma space straight b subscript 2 equals negative 1 comma space straight c subscript 2 equals 9 therefore space space straight a subscript 1 over straight a subscript 2 equals 6 over 2 equals 3 comma space straight b subscript 1 over straight b subscript 2 equals fraction numerator negative 3 over denominator negative 1 end fraction equals 3 comma space straight c subscript 1 over straight c subscript 2 equals 10 over 9 rightwards double arrow straight a subscript 1 over straight a subscript 2 equals straight b subscript 1 over straight b subscript 2 not equal to straight c subscript 1 over straight c subscript 2

Thus, the lines representing the pair of linear equations are parallel.

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