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Question
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
Solution
These equations can be written as
0x + 2y − 3z = 0
x + 3y + 0z = − 4
3x + 4y + 0z = 3
\[D = \begin{vmatrix}0 & 2 & - 3 \\ 1 & 3 & 0 \\ 3 & 4 & 0\end{vmatrix}\]
\[ = 0(0 - 0) - 2(0 - 0) - 3(4 - 9)\]
\[ = 15\]
\[ D_1 = \begin{vmatrix}0 & 2 & - 3 \\ - 4 & 3 & 0 \\ 3 & 4 & 0\end{vmatrix}\]
\[ = 0(0 - 0) - 2(0 - 0) - 3( - 16 - 9)\]
\[ = 75\]
\[ D_2 = \begin{vmatrix}0 & 0 & - 3 \\ 1 & - 4 & 0 \\ 3 & 3 & 0\end{vmatrix}\]
\[ = 0(0 - 0) - 0(0 - 0) - 3(3 + 12)\]
\[ = - 45\]
\[ D_3 = \begin{vmatrix}0 & 2 & 0 \\ 1 & 3 & - 4 \\ 3 & 4 & 3\end{vmatrix}\]
\[ = 0(9 + 16) - 2(3 + 12) - 0(4 - 9)\]
\[ = - 30\]
Now,
\[x = \frac{D_1}{D} = \frac{75}{15} = 5\]
\[y = \frac{D_2}{D} = \frac{- 45}{15} = - 3\]
\[z = \frac{D_3}{D} = \frac{- 30}{15} = - 2\]
\[ \therefore x = 5, y = - 3\text{ and }z = - 2\]
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