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Consider the system of linear equations ${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$, ${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$ and ${a_3}x + {b_3}y + {c_3}z + {d_3} = 0$. Let us denote by $\Delta (a,b,c)$ the determinant $\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ if $\Delta (a,b,c) \ne 0$, then the value of $x$ in the unique solution of the above equations is
$\frac{{\Delta (bcd)}}{{\Delta (abc)}}$
$\frac{{ - \Delta (bcd)}}{{\Delta (abc)}}$
$\frac{{\Delta (acd)}}{{\Delta (abc)}}$
$ - \frac{{\Delta (abd)}}{{\Delta (abc)}}$
Solution
(b) From Cramer's rule, $x = \frac{{{D_x}}}{D} = \frac{{\left[ {\begin{array}{*{20}{c}}{ – {d_1}}&{{b_1}}&{{c_1}}\\{ – {d_2}}&{{b_2}}&{{c_2}}\\{ – {d_3}}&{{b_3}}&{{c_3}}\end{array}} \right]}}{{\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right]}}$
$x = \frac{{ – \Delta (bcd)}}{{\Delta (abc)}}$ .
