Condition for Unique Solution of a Pair of Linear Equations

Video Explanation

Question

Obtain the condition for the following system of linear equations to have a unique solution:

\[ ax + by = c, \qquad lx + my = n \]

Solution

Step 1: Write in Standard Form

\[ ax + by – c = 0 \quad (1) \]

\[ lx + my – n = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = a, \quad b_1 = b \]

\[ a_2 = l, \quad b_2 = m \]

Step 3: Condition for a Unique Solution

A pair of linear equations has a unique solution if

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

Step 4: Apply to the Given Equations

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

Cross-multiplying,

\[ am \neq bl \]

Conclusion

The given system of linear equations has a unique solution if:

\[ \boxed{am \neq bl} \]

\[ \therefore \quad ax + by = c \text{ and } lx + my = n \text{ intersect in exactly one point.} \]