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# On comparing the ratios $${{a_1}\over {a_2}},{{b_1}\over {b_2}},{{c_1}\over {c_2}}$$ find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) $$5x - 4y + 8 = 0, 7x + 6y – 9 = 0$$ (ii)$$9x + 3y + 12 = 0, 18x + 6y + 24 = 0$$ (iii)$$6x - 3y + 10 = 0, 2x – y + 9 = 0$$

Ans.(i) $$5x - 4y + 8 = 0, 7x + 6y – 9 = 0$$

On comparing these equations with the general form: $$ax^2 + bx + c$$

$$a_1 = 5, b_1 = -4, c_1 = 8$$,
$$a_2 = 7, b_2 = 6, c_2 = -9$$

Here $${{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}$$ as,

$${{5}\over {7}} \ne {{-4}\over {6}}$$
So, these lines have a unique solution which means they intersect at one point.

(ii) $$9x + 3y + 12 = 0,18x + 6y + 24 = 0$$

On comparing these equations with the general form: $$ax^2 + bx + c$$

$$a_1 = 9, b_1 = 3, c_1 = 12$$
$$a_2 = 18, b_2 = 6, c_2 = 24$$

Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} = {{c_1}\over {c_2}}$$ as,

$${{9}\over {18}} = {{3}\over {6}} = {{12}\over {24}}$$

So, these lines are coincident.

(iii) $$6x - 3y + 10 = 0,2x – y + 9 = 0$$

On comparing these equations with the general form: $$ax^2 + bx + c$$

$$a_1 = 6, b_1 = -3, c_1 = 10$$
$$a_2 = 2, b_2 = -1, c_2 = 9$$

Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}$$ as,

$${{6}\over {2}} = {{-3}\over {-1}} \ne {{10}\over {9}}$$

So, these lines are parallel to each other.