Different Geometrical Representations of a Pair of Linear Equations

Video Explanation

Given Equation

The given linear equation is:

\[ 2x + 3y – 8 = 0 \]

Solution

(i) Intersecting Lines

For two lines to intersect, their coefficients must satisfy:

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

Choose another equation such as:

\[ x + y – 4 = 0 \]

Here,

\[ \frac{2}{1} \ne \frac{3}{1} \]

Hence, the two lines intersect.

Required pair:

\[ 2x + 3y – 8 = 0,\quad x + y – 4 = 0 \]

---

(ii) Parallel Lines

For two lines to be parallel:

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

Choose another equation such as:

\[ 4x + 6y – 10 = 0 \]

Here,

\[ \frac{2}{4} = \frac{3}{6} \ne \frac{8}{10} \]

Hence, the lines are parallel.

Required pair:

\[ 2x + 3y – 8 = 0,\quad 4x + 6y – 10 = 0 \]

---

(iii) Coincident Lines

For two lines to be coincident:

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

Choose another equation such as:

\[ 4x + 6y – 16 = 0 \]

Here,

\[ \frac{2}{4} = \frac{3}{6} = \frac{8}{16} \]

Hence, the two equations represent the same line.

Required pair:

\[ 2x + 3y – 8 = 0,\quad 4x + 6y – 16 = 0 \]

Conclusion

Thus, suitable equations for intersecting, parallel and coincident lines have been written using the given equation.