Find the ordered quintuplet \((a,b,c,d,e)\)that satisfies the system of equations

\(\begin{array}{rcrcrcrcrcr} a & + & 2b & + & 3c & + & 4d & + & 5e & = & 41, \\ 2a & + & 3b & + & 4c & + & 5d & + & e & = & 15, \\ 3a & + & 4b & + & 5c & + & d & + & 2e & = & 34, \\ 4a & + & 5b & + & c & + & 2d & + & 3e & = & 63, \\ 5a & + & b & + & 2c & + & 3d & + & 4e & = & 57. \end{array}\)

youngwolf May 6, 2023

#1**0 **

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

Guest May 6, 2023