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:

\[ 4x + 5y = 3, \qquad kx + 15y = 9 \]

Solution

Step 1: Write in Standard Form

\[ 4x + 5y – 3 = 0 \quad (1) \]

\[ kx + 15y – 9 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 4, \quad b_1 = 5, \quad c_1 = -3 \]

\[ a_2 = k, \quad b_2 = 15, \quad c_2 = -9 \]

Step 3: 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 4: Apply the Condition

\[ \frac{b_1}{b_2} = \frac{5}{15} = \frac{1}{3}, \qquad \frac{c_1}{c_2} = \frac{-3}{-9} = \frac{1}{3} \]

So,

\[ \frac{a_1}{a_2} = \frac{4}{k} = \frac{1}{3} \]

\[ k = 12 \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{k = 12} \]

\[ \therefore \quad 4x + 5y = 3 \text{ and } 12x + 15y = 9 \text{ represent the same line.} \]