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Question Number 189132 by Shrinava last updated on 12/Mar/23

$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$$$$\mathrm{in}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{equations}:$$$$\{\begin{array}{l}13\mathrm{a}+2\mathrm{b}+\mathrm{c}+6\mathrm{d}+2\mathrm{e}=96\\ 5\mathrm{a}+9\mathrm{b}+2\mathrm{c}+7\mathrm{d}+3\mathrm{e}=75\\ 7\mathrm{a}+8\mathrm{b}+17\mathrm{c}+11\mathrm{d}+7\mathrm{e}=99\\ 3\mathrm{a}+3\mathrm{b}+3\mathrm{c}+\mathrm{d}+8\mathrm{e}=55\\ \mathrm{a}+7\mathrm{b}+6\mathrm{c}+4\mathrm{d}+9\mathrm{e}=79\end{array}$$

Answered by mr W last updated on 12/Mar/23

$$sayf=a+b+c+d+e$$$$\Rightarrow a+b+c+d+e-f=0...\left(6\right)$$$$D=\left|\begin{array}{cccccc}13& 2& 1& 6& 2& 0\\ 5& 9& 2& 7& 3& 0\\ 7& 8& 17& 11& 7& 0\\ 3& 3& 3& 1& 8& 0\\ 1& 7& 6& 4& 9& 0\\ 1& 1& 1& 1& 1& -1\end{array}\right|=7047$$$$D_f=\left|\begin{array}{cccccc}13& 2& 1& 6& 2& 96\\ 5& 9& 2& 7& 3& 75\\ 7& 8& 17& 11& 7& 99\\ 3& 3& 3& 1& 8& 55\\ 1& 7& 6& 4& 9& 79\\ 1& 1& 1& 1& 1& 0\end{array}\right|=98172$$$$\Rightarrow f=a+b+c+d+e=\frac{D_f}{D}=\frac{98172}{7047}=\frac{404}{29}✓$$

Commented by manxsol last updated on 12/Mar/23

$$Ineversawthatartifice.$$$$todayisgreat!$$

Commented by mr W last updated on 13/Mar/23

$$ithinkthismethodisbetterthan$$$$solvingthe5equationsfora,b,c,d,e$$$$andthencalculatinga+b+c+d+e.$$

Commented by manxsol last updated on 13/Mar/23

$$icananyrelationbetweenthevariablesvery,verygoodSirW.$$

Commented by Shrinava last updated on 13/Mar/23

$$\mathrm{thankyou}\mathrm{dearProfessor}\mathrm{cool}$$

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