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:

\[ 2x – 3y = 7, \qquad (k+2)x – (2k+1)y = 3(2k-1) \]

Solution

Step 1: Write in Standard Form

\[ 2x – 3y – 7 = 0 \quad (1) \]

\[ (k+2)x – (2k+1)y – 3(2k-1) = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 2, \quad b_1 = -3, \quad c_1 = -7 \]

\[ a_2 = k+2, \quad b_2 = -(2k+1), \quad c_2 = -3(2k-1) \]

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

\[ \frac{2}{k+2} = \frac{3}{2k+1} \]

\[ 2(2k+1) = 3(k+2) \]

\[ 4k + 2 = 3k + 6 \]

\[ k = 4 \]

Now check with the third ratio:

\[ \frac{2}{k+2} = \frac{2}{6} = \frac{1}{3}, \qquad \frac{7}{3(2k-1)} = \frac{7}{21} = \frac{1}{3} \]

Hence, the condition is satisfied.

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{k = 4} \]

\[ \therefore \quad 2x – 3y = 7 \text{ and } 6x – 9y = 21 \text{ represent the same line.} \]