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:

\[ 2x + 3y – 5 = 0, \qquad 6x + ky – 15 = 0 \]

Solution

Step 1: Identify Coefficients

From the given equations,

\[ a_1 = 2, \quad b_1 = 3, \quad c_1 = -5 \]

\[ a_2 = 6, \quad b_2 = k, \quad c_2 = -15 \]

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

\[ \frac{a_1}{a_2} = \frac{2}{6} = \frac{1}{3}, \qquad \frac{c_1}{c_2} = \frac{-5}{-15} = \frac{1}{3} \]

So,

\[ \frac{b_1}{b_2} = \frac{3}{k} = \frac{1}{3} \]

\[ k = 9 \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{k = 9} \]

\[ \therefore \quad 2x + 3y – 5 = 0 \text{ and } 6x + 9y – 15 = 0 \text{ represent the same line.} \]