Answer to Question #114748 in Algebra for ANJU JAYACHANDRAN

Question

Use Cramer’s rule to solve
x + 2y + z =5
2x + 2y + z =6
x + 2y + 3z =9 .

Answer 1

 creating a matrix with coefficients in this equation 

$\begin{vmatrix} 1 & 2& 1&5 \\ 2 & 2& 1& 6\\ 1& 2&3 & 9 \end{vmatrix} (1) \begin{vmatrix} 1 & 2& 1 \\ 2 & 2& 1\\ 1& 2&3 \end{vmatrix} (2)$ finding determinan of matrix (2) A = -4

Replace the 1st column of the main matrix with the solution vector(5,6,9) and find its determinant

$\begin{vmatrix} 5 & 2& 1 \\ 6 & 2& 1\\ 9& 2&3 \end{vmatrix}$ B = -4

Replace the 2nd column of the main matrix with the solution vector and find its determinant

$\begin{vmatrix} 1 & 5& 1 \\ 2 & 6& 1\\ 1& 9&3 \end{vmatrix}$ C = -4

Replace the 3rd column of the main matrix with the solution vector and find its determinant

$\begin{vmatrix} 1 & 2& 5 \\ 2 & 2& 6\\ 1& 2&9 \end{vmatrix}$ D = -8

x=B/A=1

y=C/A=1

z=D/A=2

Solution set:

x=1

y=1

z=2

check

1*1+2*1+1*2 = 5

2*1+2*1+1*2 = 6

1*1+2*1+3*2 = 9