Determine the mean of the numbers a, b, c, and d. -  System of equations.?

Find:

Mean = (a + b + c + d) / 4 = ?

From the problem description:

a + (b + c + d) / 3 = x₁

b + (a + c + d) / 3 = x₂

c + (a + b + d) / 3 = x₃

d + (a + b + c) / 3 = x₄

We know the numbers x₁ through x₄, but not exactly which is which, but we do know that:

x₁ + x₂ + x₃ + x₄ = 60 + 64 + 68 + 72

Distribute:

a + (1/3)b + (1/3)c + (1/3)d = x₁

b + (1/3)a + (1/3)c + (1/3)d = x₂

c + (1/3)a + (1/3)b + (1/3)d = x₃

d + (1/3)a + (1/3)b + (1/3)c = x₄

Add all equations together:

a + (1/3)b + (1/3)c + (1/3)d + b + (1/3)a + (1/3)c + (1/3)d + c + (1/3)a + (1/3)b + (1/3)d + d + (1/3)a + (1/3)b + (1/3)c = 60 + 64 + 68 + 72

a + b + c + d + a + b + c + d = 60 + 64 + 68 + 72

2(a + b + c + d) = 60 + 64 + 68 + 72

a + b + c + d = (60 + 64 + 68 + 72) / 2

Substitute this result into the mean:

(a + b + c + d) / 4 = [(60 + 64 + 68 + 72) / 2] / 4

Mean = (60 + 64 + 68 + 72) / 8

Mean = 33

If you add "a" plus the mean of the other three, you get 60.

If you add "b" plus the mean of the other three, you get 64, etc.

So writing these as a system of four equations and four unknowns we get:

a + (b + c + d) / 3 = 60

b + (a + c + d) / 3 = 64

c + (a + b + d) / 3 = 68

d + (a + b + c) / 3 = 72

And you want to know the mean of all 4 numbers:

(a + b + c + d) / 4

Let's start out by multiplying all four equations by 3 to get rid of the fractions:

a + (b + c + d) / 3 = 60 and b + (a + c + d) / 3 = 64 and c + (a + b + d) / 3 = 68 and d + (a + b + c) / 3 = 72

3a + b + c + d = 180 and 3b + a + c + d = 192 and 3c + a + b + d = 204 and 3d + a + b + c = 216

Now I'll solve the last equation for a in terms of the other three then substitute into the other three equations:

3d + a + b + c = 216

a = 216 - b - c - 3d

3a + b + c + d = 180 and 3b + a + c + d = 192 and 3c + a + b + d = 204

3(216 - b - c - 3d) + b + c + d = 180 and 3b + 216 - b - c - 3d + c + d = 192 and 3c + 216 - b - c - 3d + b + d = 204

Now simplify:

648 - 3b - 3c - 9d + b + c + d = 180 and 2b - 2d = -24 and 2c - 2d = -12

We actually had additional terms cancel out in the second and third equations which make things easier.  We can continue to simplify the last two by dividing both sides by 2 as we simplify the first one further:

-2b - 2c - 8d = -468 and b - d = -12 and c - d = -6

I'll simplify the first equation by dividing both sides by -2 and solve the second equation for b in terms of d:

b + c + 4d = 234 and b = d - 12 and c - d = -6

Now we can substitute the expression for b in terms of d into the other two equations:

b + c + 4d = 234 and c - d = -6

d - 12 + c + 4d = 234 and c - d = -6

c + 5d = 246 and c - d = -6

Solve the last equation for c in terms of d and substitute again:

c - d = -6

c = d - 6

c + 5d = 246

d - 6 + 5d = 246

6d = 252

d = 42

Now that we know d we can start working back and solve for the others:

b = d - 12 and c = d - 6

b = 42 - 12 and c = 42 - 6

b = 30 and c = 36

a = 216 - b - c - 3d

a = 216 - 30 - 36 - 3(42)

a = 216 - 30 - 36 - 126

a = 24

Now we have the four numbers we can find the mean:

(a + b + c + d) / 4

(24 + 30 + 36 + 42) / 4

132 / 4

33 is the mean of your four numbers.