Browse Questions

# Solve the following non-homogeneous system of linear equation by determinant method : $3x+2y=5\;,x+3y=4$

Toolbox:
• Cramer's rule : If $Ax = B$ is a system of x linear equations in x unknowns such that $|A| \neq 0$ then the system has a unique solution. Let the x unknowns be $x_1, x_2.........x_n$ and $\Delta$ denote $|A|$. The solution is given by $x_1=\large\frac{\Delta x_1}{\Delta}, x_2=\large\frac{\Delta x_2}{\Delta}..........x_n = \large\frac{\Delta x_n}{\Delta}$ where $\Delta x_1, \Delta x_2....\Delta x_n$ are obtained by replacing the 1st, 2nd.....$n^{th}$ column respectively by the column of constants in B.
Step 1
$\Delta = \begin{vmatrix} 3 & 2 \\ 1 & 3 \end{vmatrix} = 9-2=7 \neq 0. \therefore$ Cramer's rule can be used.
Step 2
$\Delta x = \begin{vmatrix} 5 & 2 \\ 4 & 3 \end{vmatrix}=15-8=7$
$\Delta y = \begin{vmatrix} 3 & 5 \\ 1 & 4 \end{vmatrix}=12-5=7$
Step 3
$x = \large\frac{\Delta x}{\Delta} = \large\frac{7}{7}=1\: \: \: y = \large\frac{\Delta y}{\Delta} = \large\frac{7}{7}=1$
$(x,y) = (1,1)$