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:

\[ x + (k+1)y = 4, \qquad (k+1)x + 9y = 5k + 2 \]

Solution

Step 1: Write in Standard Form

\[ x + (k+1)y – 4 = 0 \quad (1) \]

\[ (k+1)x + 9y – (5k+2) = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 1, \quad b_1 = k+1, \quad c_1 = -4 \]

\[ a_2 = k+1, \quad b_2 = 9, \quad c_2 = -(5k+2) \]

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

First compare the first two ratios:

\[ \frac{1}{k+1} = \frac{k+1}{9} \]

\[ (k+1)^2 = 9 \]

\[ k+1 = \pm 3 \]

\[ k = 2 \quad \text{or} \quad k = -4 \]

Now check with the third ratio:

For \(k = 2\),

\[ \frac{1}{3} = \frac{3}{9} = \frac{4}{12} \]

Condition is satisfied.

For \(k = -4\),

\[ \frac{1}{-3} \neq \frac{4}{-18} \]

Condition is not satisfied.

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{k = 2} \]

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