Consistency of a Pair of Linear Equations

Video Explanation

Question

Find the value of \(k\) for which the system of equations

\[ kx + 2y = 5, \qquad 3x + y = 1 \]

has (i) a unique solution (ii) no solution.

Solution

Step 1: Write in Standard Form

\[ kx + 2y – 5 = 0 \quad (1) \]

\[ 3x + y – 1 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

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

\[ a_2 = 3, \quad b_2 = 1, \quad c_2 = -1 \]

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(i) Condition for a Unique Solution

A pair of linear equations has a unique solution if

\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

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

\[ k \neq 6 \]

Hence, the system has a unique solution for all real values of \(k\) except \(k = 6\).

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(ii) Condition for No Solution

A pair of linear equations has no solution if

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

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

\[ k = 6 \]

Now check the third ratio:

\[ \frac{c_1}{c_2} = \frac{-5}{-1} = 5 \]

Since

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

the system is inconsistent.

Conclusion

(i) The system has a unique solution for:

\[ \boxed{k \neq 6} \]

(ii) The system has no solution for:

\[ \boxed{k = 6} \]