Question

What is the condition for a unique solution of a pair of linear equations in two variables?

Answer 1

Hint: In this question first assume any pair of linear equations in two variables and convert them into matrix format so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Let us consider the system of linear equations in two variables be
$
  ax + by = c \\
  dx + ey = f \\
$
Where (x, y) are the variables and (a, b, c, d, e, f) are the constants.
Now convert this system of equation into matrix format we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  a&b \\
  d&e
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  c \\
  f
\end{array}} \right]$
In above matrix format determinant (D) = $\left| {\begin{array}{*{20}{c}}
  a&b \\
  d&e
\end{array}} \right|$
So, the condition of unique solution of a pair of linear equation in two variables is
The value of determinant (D) should not equal zero.
$ \Rightarrow D \ne 0$
Or,
$D = \left| {\begin{array}{*{20}{c}}
  a&b \\
  d&e
\end{array}} \right| \ne 0$
Expand the determinant we have
$D = \left( {ae - bd} \right) \ne 0$
So, this is the required condition of a unique solution of a pair of linear equations in two variables.
So, this is the required answer.

Note: In such types of questions first let any two linear equations as above then convert it into matrix format as above and calculate the value of determinate so, for unique solution the value of determinant should not equal to zero if zero then the system of equations has either no solution or infinitely many solutions.