Condition for No Solution of a Pair of Linear Equations

Video Explanation

Question

Find the value of \(k\) for which the following system of equations has no solution:

\[ kx – 5y = 2, \qquad 6x + 2y = 7 \]

Solution

Step 1: Write in Standard Form

\[ kx – 5y – 2 = 0 \quad (1) \]

\[ 6x + 2y – 7 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

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

\[ a_2 = 6, \quad b_2 = 2, \quad c_2 = -7 \]

Step 3: Condition for No Solution

A pair of linear equations has no solution if

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

Step 4: Apply the Condition

\[ \frac{b_1}{b_2} = \frac{-5}{2} \]

So,

\[ \frac{k}{6} = \frac{-5}{2} \]

\[ k = -15 \]

Now check the third ratio:

\[ \frac{c_1}{c_2} = \frac{-2}{-7} = \frac{2}{7} \]

Since

\[ \frac{k}{6} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}, \]

the system is inconsistent.

Conclusion

The given system of equations has no solution for:

\[ \boxed{k = -15} \]

\[ \therefore \quad -15x – 5y = 2 \text{ and } 6x + 2y = 7 \text{ represent parallel lines.} \]