2x – 3y = 0 and 2x + αy = 0

For what value of α the system has unique solution.

Concept:

The system of equations A X = 0 is said to be homogenous system of equations, then

If |A| ≠ 0, then its solution X = 0, is called trivial solution.

If |A| = 0. Then A X = 0 has a non-trivial solution which means the system will have infinitely many solutions.

Calculation:

Given:  2x – 3y = 0 and 2x + αy = 0

These equations can be written as: A X = B where \(A = \left[ {\begin{array}{*{20}{c}} 2&{ - 3}\\ 2&\alpha \end{array}} \right],\;X = \left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]\;and\;B = \left[ {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right]\)

As we know that, the given system is a homogenous system of equation. So, in order to say that the system has unique solution: |A| ≠ 0.

⇒ |A| = 2α + 6 ≠ 0 ⇒ α ≠ -3.