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Mathematical Methods for Economics — Economics STD 11 — Question

Tamilnadu BoardEnglish MediumSTD 11EconomicsMathematical Methods for Economics2 Marks
Question
Find the solution of the equation system
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Answer

$7 x_1-x_2-x_3=0$
$10 x_1-2 x_2+x_3=8$
$6 x_1+3 x_2-2 x_3=7$
The matrix form is
$\left[\begin{array}{ccc}7 & -1 & -1 \\ 10 & -2 & 1 \\ 6 & 3 & -2\end{array}\right]\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right]=\left[\begin{array}{l}0 \\ 8 \\ 7\end{array}\right]$
$\Delta=\left|\begin{array}{ccc}7 & -1 & -1 \\ 10 & -2 & 1 \\ 6 & 3 & -2\end{array}\right|$
$= 7 (4 – 3) + 1 (-20 -6) -1 (30 + 12)$
$= 7 (1) + 1 (-26) -1 (42)$
$+ 7 -26 – 42$
$\triangle = – 61$
$x_1=\left|\begin{array}{ccc}0 & -1 & -1 \\ 8 & -2 & 1 \\ 7 & 3 & -2\end{array}\right|$
$= 0 (4 -3) + 1 (-16 -7) – 1 (24+14)$
$= 0 + 1 (-23) – 1 (38)$
$= -23 – 38$
$= -61$
$x_2=\left|\begin{array}{ccc}7 & 0 & -1 \\ 10 & 8 & 1 \\ 6 & 7 & -2\end{array}\right|$
$= 7 (-16 – 7) – 0 (- 20 – 6) – 1 (70 – 48)$
$= 7 ( -23) + 0 (-20 – 6) – 1(70 – 48)$
$= – 161 – 22$
$= – 183$
$x_3=\left|\begin{array}{ccc}7 & -1 & 0 \\ 10 & -2 & 8 \\ 6 & 3 & 7\end{array}\right|$
$= 7(-14 -24) + 1 (70 – 48) + 1 (30 + 12)$
$= 7 (-38) + 1 (22) + 0$
$= – 266 + 22$
$= – 244$
$x_1=\frac{\Delta x_1}{\Delta}=\frac{-61}{-61}=1$$x_2=\frac{\Delta x_2}{\Delta}=\frac{-183}{-61}=3$$x_3=\frac{\Delta x_3}{\Delta}=\frac{-244}{-61}=4$
$x_1, x_2, x_3=1,3,4$

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