Answer:
a = 23
b = 31
c = 60
d = 33
e = 51
Step-by-step explanation:
Given that the sum of the numbers in EACH of the four ROWS is the SAME, then:
[tex]75 + b + 83 = 76 + 80 + d = a + 81 + 85 = 78 + c + e[/tex]
This simplifies to:
[tex]\textsf{Equation 1:}\quad b+158 = d+156= a + 166 = c + e+78[/tex]
Express a in terms of b, and express d in terms of b:
[tex]a + 166 = b + 158\\\\a = b - 8[/tex]
[tex]d + 156 = b + 158\\\\d = b + 2[/tex]
Given that the sum of the numbers in EACH of the three COLUMNS is the SAME, then:
[tex]75+76+a+78=b+80+81+c=83+d+85+e[/tex]
This simplifies to:
[tex]\textsf{Equation 2:}\quad a+229=b+c+161=d+e+168[/tex]
Substitute in a = b - 8 and d = b + 2 to equation 2:
[tex]b - 8+229=b+c+161=b + 2 +e+168\\\\b+221=b+c+161=b+e+170[/tex]
As b + 221 = b + c + 161 then we can solve for c:
[tex]b + 221 = b + c + 161\\\\221=c+161\\\\\boxed{\boxed{c=60}}[/tex]
Substitute c = 60 into the original two equations:
[tex]\textsf{Equation 1:}\quad b+158 = d+156= a + 166 = e+138[/tex]
[tex]\textsf{Equation 2:}\quad a+229=b+221=d+e+168[/tex]
Using equation 1, express b, d and e in terms of a:
[tex]b+158=a+166\\\\b=a+8[/tex]
[tex]d+156=a+166\\\\d=a+10[/tex]
[tex]e+138=a+166\\\\e=a+28[/tex]
Substitute these expressions into equation 2 and solve for a:
[tex]a+229=a+8+221=a+10+a+28+168\\\\a+229=2a+206\\\\a+206=229\\\\\boxed{\boxed{a=23}}[/tex]
Now, substitute a = 23 into the expressions for b, d and e in terms of a:
[tex]b=23+8\\\\\boxed{\boxed{b=31}}[/tex]
[tex]d=23+10\\\\\boxed{\boxed{d=33}}[/tex]
[tex]e=23+28\\\\\boxed{\boxed{e=51}}[/tex]
Therefore, the values of a, b, c, d and e are:
- a = 23
- b = 31
- c = 60
- d = 33
- e = 51
