# Solve the following non-homogeneous system of linear equation by determinant method : $4x+5y=9\;,8x+10y=18$

Toolbox:
• Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $\Delta \neq 0$ thesystem is consistent and has a unique solution. Cramer's rule can be used. Case 2 : If $\Delta = 0$, there are two / ( four) possibilities:
• 2a: If $\Delta = 0$, and atleast one of the values $\Delta_ x, \Delta_ y ( or \: \Delta_ z )$ is nonzero, the system is inconsistent and has no solution.
• 2b: If $\Delta = 0\: and \: \Delta_x=\Delta_y ( = \Delta_z)=0$ and at least one element ( second order minor ) of $\Delta$ is nonzero, the system is consistent with infinitely many solution. The system reduces to one equation ( two equations )
• 2c: If $\Delta = 0$ and $\Delta_x=\Delta_y = \Delta_z=0$ and all their 2 x 2 minors are 0, and $\Delta$ has atleast one nonzero element, then the system is consistent with infinitely many solutions. The system reduces to one equation.
• 2d: If $\Delta = 0, \Delta_x=\Delta_y = \Delta_z=0$, all 2 x 2 minors of $\Delta$ are zero but there is at least one 2 x 2 minor of $\Delta_x, \Delta_y \: or \: \Delta_z$ that is nonzero, then the system is inconsistent and has no solution.
Step 1
$\Delta = \begin{vmatrix} 4 & 5 \\ 8 & 10 \end{vmatrix} = 40-40=0$
The system may or maynot be consistent.
Step 2
$\Delta_x = \begin{vmatrix} 9 & 5 \\ 18 & 10 \end{vmatrix} = 90-90=0$
$\Delta_y = \begin{vmatrix} 4 & 9 \\ 8 & 18 \end{vmatrix} = 72-72=0$
$\Delta = \Delta_x=\Delta_y=0$ and $\Delta$ has nonzero elements. $\therefore$ the system is consistent with infinitely many solutions. It reduces to a single equation.
$4x+5y=9$
Step 3
Let $x = t, t \in R$
Then $y=\large\frac{9-4t}{5}$
The solution set is $(x,y) = \bigg(t, \large\frac{9-4t}{5} \bigg), t \in R$