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Find-the-value-of-a-b-c-d-e-in-the-system-of-equations-13a-2b-c-6d-2e-96-5a-9b-2c-7d-3e-75-7a-8b-17c-11d-7e-99-3a-3b-3c-d-8e-55-a-7b-6c-4d-9e-79-




Question Number 189132 by Shrinava last updated on 12/Mar/23
Find the value of   a+b+c+d+e  in the system of equations:   { ((13a+2b+c+6d+2e=96)),((5a+9b+2c+7d+3e=75)),((7a+8b+17c+11d+7e=99)),((3a+3b+3c+d+8e=55)),((a+7b+6c+4d+9e=79)) :}
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}: \\ $$$$\begin{cases}{\mathrm{13a}+\mathrm{2b}+\mathrm{c}+\mathrm{6d}+\mathrm{2e}=\mathrm{96}}\\{\mathrm{5a}+\mathrm{9b}+\mathrm{2c}+\mathrm{7d}+\mathrm{3e}=\mathrm{75}}\\{\mathrm{7a}+\mathrm{8b}+\mathrm{17c}+\mathrm{11d}+\mathrm{7e}=\mathrm{99}}\\{\mathrm{3a}+\mathrm{3b}+\mathrm{3c}+\mathrm{d}+\mathrm{8e}=\mathrm{55}}\\{\mathrm{a}+\mathrm{7b}+\mathrm{6c}+\mathrm{4d}+\mathrm{9e}=\mathrm{79}}\end{cases} \\ $$
Answered by mr W last updated on 12/Mar/23
say f=a+b+c+d+e  ⇒ a+b+c+d+e−f=0     ...(6)  D= determinant (((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)))=7047  D_f = determinant (((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))=98172  ⇒f=a+b+c+d+e=(D_f /D)=((98172)/(7047))=((404)/(29)) ✓
$${say}\:{f}={a}+{b}+{c}+{d}+{e} \\ $$$$\Rightarrow\:{a}+{b}+{c}+{d}+{e}−{f}=\mathrm{0}\:\:\:\:\:…\left(\mathrm{6}\right) \\ $$$${D}=\begin{vmatrix}{\mathrm{13}}&{\mathrm{2}}&{\mathrm{1}}&{\mathrm{6}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{5}}&{\mathrm{9}}&{\mathrm{2}}&{\mathrm{7}}&{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{7}}&{\mathrm{8}}&{\mathrm{17}}&{\mathrm{11}}&{\mathrm{7}}&{\mathrm{0}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{3}}&{\mathrm{1}}&{\mathrm{8}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{7}}&{\mathrm{6}}&{\mathrm{4}}&{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{1}}\end{vmatrix}=\mathrm{7047} \\ $$$${D}_{{f}} =\begin{vmatrix}{\mathrm{13}}&{\mathrm{2}}&{\mathrm{1}}&{\mathrm{6}}&{\mathrm{2}}&{\mathrm{96}}\\{\mathrm{5}}&{\mathrm{9}}&{\mathrm{2}}&{\mathrm{7}}&{\mathrm{3}}&{\mathrm{75}}\\{\mathrm{7}}&{\mathrm{8}}&{\mathrm{17}}&{\mathrm{11}}&{\mathrm{7}}&{\mathrm{99}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{3}}&{\mathrm{1}}&{\mathrm{8}}&{\mathrm{55}}\\{\mathrm{1}}&{\mathrm{7}}&{\mathrm{6}}&{\mathrm{4}}&{\mathrm{9}}&{\mathrm{79}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{0}}\end{vmatrix}=\mathrm{98172} \\ $$$$\Rightarrow{f}={a}+{b}+{c}+{d}+{e}=\frac{{D}_{{f}} }{{D}}=\frac{\mathrm{98172}}{\mathrm{7047}}=\frac{\mathrm{404}}{\mathrm{29}}\:\checkmark \\ $$
Commented by manxsol last updated on 12/Mar/23
I never saw that artifice.  today is great!
$${I}\:{never}\:{saw}\:{that}\:{artifice}. \\ $$$${today}\:{is}\:{great}! \\ $$
Commented by mr W last updated on 13/Mar/23
i think this method is better than  solving the 5 equations for a,b,c,d,e  and then calculating a+b+c+d+e.
$${i}\:{think}\:{this}\:{method}\:{is}\:{better}\:{than} \\ $$$${solving}\:{the}\:\mathrm{5}\:{equations}\:{for}\:{a},{b},{c},{d},{e} \\ $$$${and}\:{then}\:{calculating}\:{a}+{b}+{c}+{d}+{e}. \\ $$
Commented by manxsol last updated on 13/Mar/23
i can any relation between the variables very,very good Sir W.
$${i}\:{can}\:{any}\:{relation}\:{between}\:{the}\:{variables}\:{very},{very}\:{good}\:{Sir}\:{W}. \\ $$
Commented by Shrinava last updated on 13/Mar/23
thankyou dearProfessor cool
$$\mathrm{thankyou}\:\mathrm{dearProfessor}\:\mathrm{cool} \\ $$

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