Condition for Unique Solution

Video Explanation

Question

If \(am \ne bl\), then the system of equations \(ax + by = c\) and \(lx + my = n\) has:

(a) a unique solution (b) no solution (c) infinitely many solutions (d) may or may not have a solution

Solution

Step 1: Identify Coefficients

For equation (1): \(a_1 = a,\; b_1 = b\)

For equation (2): \(a_2 = l,\; b_2 = m\)

Step 2: Apply Condition

We are given:

\[ am \ne bl \]

Divide both sides by \(al\) (assuming non-zero):

\[ \frac{a}{l} \ne \frac{b}{m} \]

Step 3: Interpretation

Since:

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

The system has a unique solution.

Final Answer

\[ \text{The system has a unique solution.} \]