Condition for Infinitely Many Solutions of a Pair of Linear Equations

Video Explanation

Question

Find the value of \(k\) for which the following system of equations has infinitely many solutions:

\[ kx – 2y + 6 = 0, \qquad 4x – 3y + 9 = 0 \]

Solution

Step 1: Identify Coefficients

From the given equations,

\[ a_1 = k, \quad b_1 = -2, \quad c_1 = 6 \]

\[ a_2 = 4, \quad b_2 = -3, \quad c_2 = 9 \]

Step 2: Condition for Infinitely Many Solutions

A pair of linear equations has infinitely many solutions if

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

Step 3: Apply the Condition

\[ \frac{b_1}{b_2} = \frac{-2}{-3} = \frac{2}{3}, \qquad \frac{c_1}{c_2} = \frac{6}{9} = \frac{2}{3} \]

So,

\[ \frac{a_1}{a_2} = \frac{k}{4} = \frac{2}{3} \]

\[ k = \frac{8}{3} \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{k = \dfrac{8}{3}} \]

\[ \therefore \quad \frac{8}{3}x – 2y + 6 = 0 \text{ and } 4x – 3y + 9 = 0 \text{ represent the same line.} \]