Pair of Linear Equations – Nature of Lines

Video Explanation

Question

Given the linear equation

\[ 2x + 3y – 8 = 0, \]

write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines
(ii) parallel lines
(iii) coincident lines

Solution

Given Equation

\[ 2x + 3y – 8 = 0 \]

Here,

\[ a_1 = 2,\quad b_1 = 3,\quad c_1 = -8 \]

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(i) Intersecting Lines

For intersecting lines:

\[ \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \]

Choose another equation:

\[ x + y – 5 = 0 \]

Here,

\[ a_2 = 1,\quad b_2 = 1 \]

\[ \frac{a_1}{a_2} = \frac{2}{1} = 2, \quad \frac{b_1}{b_2} = \frac{3}{1} = 3 \]

Since

\[ \frac{a_1}{a_2} \ne \frac{b_1}{b_2}, \]

the lines intersect at a point.

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(ii) Parallel Lines

For parallel lines:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \]

Choose another equation:

\[ 4x + 6y – 10 = 0 \]

Here,

\[ a_2 = 4,\quad b_2 = 6,\quad c_2 = -10 \]

\[ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \]

but

\[ \frac{c_1}{c_2} = \frac{-8}{-10} = \frac{4}{5} \]

Since

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}, \]

the lines are parallel.

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(iii) Coincident Lines

For coincident lines:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Choose another equation:

\[ 4x + 6y – 16 = 0 \]

Here,

\[ a_2 = 4,\quad b_2 = 6,\quad c_2 = -16 \]

\[ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{-8}{-16} = \frac{1}{2} \]

Since

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, \]

the lines coincide.

Conclusion

(i) Intersecting lines: \(x + y – 5 = 0\)
(ii) Parallel lines: \(4x + 6y – 10 = 0\)
(iii) Coincident lines: \(4x + 6y – 16 = 0\)