Find the ordered quintuplet $(a,b,c,d,e)$ that satisfies the system of equations\[ \begin{array}{rcrcrcrcrcr} a & + & 2b & + & 3c & + & 4d & + & 5e & = & 41, \\ 2a & + & 3b & + & 4c & + & 5d & + & e & = & 15, \\ 3a & + & 4b & + & 5c & + & d & + & 2e & = & 34, \\ 4a & + & 5b & + & c & + & 2d & + & 3e & = & 63, \\ 5a & + & b & + & 2c & + & 3d & + & 4e & = & 57. \end{array} \]
This is a tricky system! But not too bad, if you take advantage of the symmetry.
The system of equations is as follows:
a + 2b + 3c + 4d + 5e = 41
2a + 3b + 4c + 5d + e = 15
3a + 4b + 5c + 1d + 2e = 34
4a + 5b + 1c + 2d + 3e = 63
5a + 1b + 2c + 3d + 4e = 57
We can solve this system of equations using the following steps:
Add the first, second, and third equations together. This gives us the following equation:
10a + 10b + 10c + 10d + 10e = 155
Divide both sides of this equation by 10. This gives us the following equation:
a + b + c + d + e = 15.5
Subtract the second equation from the first equation. This gives us the following equation:
a + b + c + d + e = 26
Subtract the third equation from the second equation. This gives us the following equation:
a + b + c + d + e = 19
Subtract the fourth equation from the third equation. This gives us the following equation:
a + b + c + d + e = 29
Subtract the fifth equation from the fourth equation. This gives us the following equation:
a + b + c + d + e = 18
Add all five equations together. This gives us the following equation:
10a + 10b + 10c + 10d + 10e = 127
Divide both sides of this equation by 10. This gives us the following equation:
a + b + c + d + e = 12.7
Now that we know the value of e, we can solve for the other variables. Substituting e = 12.7 into the first equation, we get the following equation:
a + 2b + 3c + 4d + 5(12.7) = 41
Solving for a, we get the following value:
a = -1.8
Substituting e = 12.7 and a = -1.8 into the second equation, we get the following equation:
2(-1.8) + 3b + 4c + 5(12.7) = 15
Solving for b, we get the following value:
b = 1.2
Substituting e = 12.7, a = -1.8, and b = 1.2 into the third equation, we get the following equation:
3(-1.8) + 4c + 5(12.7) = 34
Solving for c, we get the following value:
c = 0.9
Substituting e = 12.7, a = -1.8, b = 1.2, and c = 0.9 into the fourth equation, we get the following equation:
4(-1.8) + 5d + 1(12.7) = 63
Solving for d, we get the following value:
d = 11.5
Therefore, the solution to the system of equations is as follows:
a = -1.8, b = 1.2, c = 0.9, d = 11.5, e = 12.7
a + 2 b + 3 c + 4 d + 5 e = 41
2 a + 3 b + 4 c + 5 d + e = 15
3 a + 4 b + 5 c + 1 d + 2 e = 34
4 a + 5 b + 1 c + 2 d + 3 e = 63
5 a + 1 b + 2 c + 3 d + 4 e = 57, solve for a, b, c, d, e
Use eliminations and substitutions to get:
a = 6 and b = 4 and c = -3 and d = -1 and e = 8 - which balance the 5 equations
