Condition for Coincident Lines

Video Explanation

Question

For what value of \(k\), will the following system of equations represent coincident lines?

\[ x + 2y + 7 = 0, \qquad 2x + ky + 14 = 0 \]

Solution

Step 1: Identify Coefficients

From the given equations,

\[ a_1 = 1, \quad b_1 = 2, \quad c_1 = 7 \]

\[ a_2 = 2, \quad b_2 = k, \quad c_2 = 14 \]

Step 2: Condition for Coincident Lines

Two linear equations represent coincident lines if

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

Step 3: Apply the Condition

\[ \frac{a_1}{a_2} = \frac{1}{2}, \qquad \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \]

So,

\[ \frac{b_1}{b_2} = \frac{2}{k} = \frac{1}{2} \]

\[ k = 4 \]

Conclusion

The given system of equations represents coincident lines for:

\[ \boxed{k = 4} \]

\[ \therefore \quad x + 2y + 7 = 0 \text{ and } 2x + 4y + 14 = 0 \text{ represent the same line.} \]