Finding Value of k for Infinite Solutions

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(2x + 3y = 5\) and \(4x + ky = 10\) has infinitely many solutions.

Solution

Step 1: Write in Standard Form

\[ 2x + 3y – 5 = 0 \]

\[ 4x + ky – 10 = 0 \]

Step 2: Identify Coefficients

For equation (1): \(a_1 = 2,\; b_1 = 3,\; c_1 = -5\)

For equation (2): \(a_2 = 4,\; b_2 = k,\; c_2 = -10\)

Step 3: Apply Condition for Infinite Solutions

For infinitely many solutions:

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

\[ \frac{2}{4} = \frac{3}{k} = \frac{-5}{-10} \]

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

Step 4: Solve

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

\[ k = 6 \]

Final Answer

\[ \text{The system has infinitely many solutions when } k = 6. \]

Verification Insight

If \(k = 6\), all ratios are equal:

\[ \frac{2}{4} = \frac{3}{6} = \frac{5}{10} = \frac{1}{2} \]

⇒ Lines coincide ⇒ Infinite solutions ✔