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Question
x+ y = 5
y + z = 3
x + z = 4
Solution
These equations can be written as
x + y + 0z = 5
0x + y + z = 3
x + 0y + z = 4
\[D = \begin{vmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{vmatrix}\]
\[ = 1(1 - 0) - 1(0 - 1) + 0(0 - 1)\]
\[ = 1(1) - 1( - 1) + 0\]
\[ = 2\]
\[ D_1 = \begin{vmatrix}5 & 1 & 0 \\ 3 & 1 & 1 \\ 4 & 0 & 1\end{vmatrix}\]
\[ = 5(1 - 0) - 1(3 - 4) + 0(0 - 4)\]
\[ = 5(1) - 1( - 1)\]
\[ = 6\]
\[ D_2 = \begin{vmatrix}1 & 5 & 0 \\ 0 & 3 & 1 \\ 1 & 4 & 1\end{vmatrix}\]
\[ = 1(3 - 4) - 5(0 - 1) + 0(0 - 4)\]
\[ = 1( - 1) - 5( - 1)\]
\[ = 4\]
\[ D_3 = \begin{vmatrix}1 & 1 & 5 \\ 0 & 1 & 3 \\ 1 & 0 & 4\end{vmatrix}\]
\[ = 1(4 - 0) - 1(0 - 3) + 5(0 - 1)\]
\[ = 1(4) - 1( - 3) + 5( - 1)\]
\[ = 2\]
Now,
\[x = \frac{D_1}{D} = \frac{6}{2} = 3\]
\[y = \frac{D_2}{D} = \frac{4}{2} = 2\]
\[z = \frac{D_3}{D} = \frac{2}{2} = 1\]
\[ \therefore x = 3, y = 2\text{ and }z = 1\]
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