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.

5x - 4y + 8 = 0, 7x + 6y - 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 5 comma space space straight b subscript 1 equals negative 4 comma space space straight c subscript 1 equals negative 8 straight a subscript 2 equals 7 comma space space straight b subscript 2 equals 6. space space straight c subscript 2 equals negative 9 therefore space space space straight a subscript 1 over straight a subscript 2 space equals space 5 over 7 comma space space straight b subscript 1 over straight b subscript 2 equals fraction numerator negative 4 over denominator 6 end fraction rightwards double arrow straight a subscript 1 over straight a subscript 2 not equal to straight b subscript 1 over straight b subscript 2

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

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