Nature of Pair of Linear Equations Using Ratio of Coefficients

Video Explanation

Question

On comparing the ratios \[ \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2}, \] find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide (without drawing them):

(i) \(5x – 4y + 8 = 0,\; 7x + 6y – 9 = 0\)
(ii) \(9x + 3y + 12 = 0,\; 18x + 6y + 24 = 0\)
(iii) \(6x – 3y + 10 = 0,\; 2x – y + 9 = 0\)

Solution

(i) \(5x – 4y + 8 = 0\) and \(7x + 6y – 9 = 0\)

\[ a_1 = 5,\; b_1 = -4,\; c_1 = 8 \]

\[ a_2 = 7,\; b_2 = 6,\; c_2 = -9 \]

\[ \frac{a_1}{a_2} = \frac{5}{7}, \quad \frac{b_1}{b_2} = \frac{-4}{6} = -\frac{2}{3} \]

Since \[ \frac{a_1}{a_2} \ne \frac{b_1}{b_2}, \] the lines intersect at a point.

(ii) \(9x + 3y + 12 = 0\) and \(18x + 6y + 24 = 0\)

\[ a_1 = 9,\; b_1 = 3,\; c_1 = 12 \]

\[ a_2 = 18,\; b_2 = 6,\; c_2 = 24 \]

\[ \frac{a_1}{a_2} = \frac{9}{18} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{12}{24} = \frac{1}{2} \]

Since \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, \] the lines coincide.

(iii) \(6x – 3y + 10 = 0\) and \(2x – y + 9 = 0\)

\[ a_1 = 6,\; b_1 = -3,\; c_1 = 10 \]

\[ a_2 = 2,\; b_2 = -1,\; c_2 = 9 \]

\[ \frac{a_1}{a_2} = \frac{6}{2} = 3, \quad \frac{b_1}{b_2} = \frac{-3}{-1} = 3, \quad \frac{c_1}{c_2} = \frac{10}{9} \]

Since \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}, \] the lines are parallel.

Conclusion

(i) Lines intersect at a point
(ii) Lines coincide
(iii) Lines are parallel