[tex]\bold{\huge{\orange{\underline{ Solution }}}}[/tex]
[tex]\bold{\underline{ Given \: Rules}}[/tex]
- The sum of the number in each of the four rows is the same
- The sum of the numbers in each of the three columns is the same
- The sum of any row does not equal the sum of any column
[tex]\bold{\underline{ Let's \: Begin}}[/tex]
According to the Second rule :-
[tex]\sf{ 75+b+83=76+80+d=a+81+85+78+c+e }[/tex]
[tex]\sf{ 158 + b = 156 + d = 166 + a = 78 + c + e ...(1)}[/tex]
According to the first rule :-
[tex]\sf{ 75+76+a+78 = b+80+81+c = 83+86+d+e}[/tex]
[tex]\sf{ 229 + a = 161 + b + c = 168 + d + e ...(2)}[/tex]
From ( 1 ) we got :-
[tex]\sf{ 158 + b = 166 + a, 156 + d = 166 + a }[/tex]
[tex]\sf{ b = 166 - 158 + a, d = 166 - 156 + a }[/tex]
[tex]\sf{ b = 8 + a, d = 10 + a ...(3)}[/tex]
Subsitute (3) in ( 2 ) :-
[tex]\sf{229+a = 161+8+a+c = 168+10+a+e}[/tex]
[tex]\sf{ 229+a = 169+a+c = 178+a+e}[/tex]
We can write it as :-
[tex]\sf{ 229+a = 169+a+c \:or\:229+a = 178+a+e}[/tex]
[tex]\sf{ c = 299-169+a-a\:or\:e = 229-178+a-a}[/tex]
[tex]\sf{ c = 60 \: and \: e = 51 }[/tex]
Subsitute the value of c and e in ( 1 ):-
[tex]\sf{ 158 + b = 156 + d = 166 + a = 78 + 60 + 51 }[/tex]
[tex]\sf{ 158 + b = 156 + d = 166 + a = 189}[/tex]
Now,
[tex]\sf{ For \: b, 158 + b = 189 }[/tex]
[tex]\sf{ b = 189 - 158 }[/tex]
[tex]\sf{ b = 31}[/tex]
[tex]\sf{ For \: d , 156 + b = 189 }[/tex]
[tex]\sf{ d = 189 - 156 }[/tex]
[tex]\sf{ d = 33}[/tex]
[tex]\sf{ For \: a, 166 + a = 189 }[/tex]
[tex]\sf{ a = 189 - 166 }[/tex]
[tex]\sf{ a = 23 }[/tex]
Hence, The value of a, b, c, d and e is 23, 31 ,60 ,33 and 51 .
