Tìm x; y và z thỏa mãn

To solve this system of equations, let's start by simplifying each equation:

1) X/Y + Z - 3 = Y/X + Z
2) Y/X + Z = Z/(X + Y + 3)
3) Z/(X + Y + 3) = 1/12(X + Y + Z)

Let's solve for X in terms of Y and Z in the first equation:

X/Y + Z - 3 = Y/X + Z
Multiply both sides by XY to eliminate the denominators:
X^2 + XYZ - 3XY = Y^2 + XZ
Rearrange the equation:
X^2 - Y^2 = 3XY - XZ - XYZ
Factor out X on the right side:
X^2 - Y^2 = X(3Y - Z - YZ)
Divide both sides by (3Y - Z - YZ):
X = (X^2 - Y^2) / (3Y - Z - YZ)

Now, let's solve for Y in terms of X and Z in the second equation:

Y/X + Z = Z/(X + Y + 3)
Multiply both sides by X(X + Y + 3) to eliminate the denominators:
Y(X + Y + 3) + XZ(X + Y + 3) = Z(X + Y + 3)
Expand the equation:
XY + Y^2 + 3Y + X^2Z + XYZ + 3XZ = ZX + ZY + 3Z
Rearrange the equation:
Y^2 + (XZ + XYZ - ZY) + (3Y - 3Z) + (X^2Z - ZX - 3XZ) = 0
Factor out Y on the left side:
Y^2 + Y(XZ + XYZ - Z) + (3Y - 3Z) + (X^2Z - ZX - 3XZ) = 0
Divide both sides by (XZ + XYZ - Z):
Y = -(3Y - 3Z + X^2Z - ZX - 3XZ) / (XZ + XYZ - Z)

Finally, let's solve for Z in terms of X and Y in the third equation:

Z/(X + Y + 3) = 1/12(X + Y + Z)
Multiply both sides by 12(X + Y + 3) to eliminate the denominators:
12Z = X + Y + Z
Rearrange the equation:
11Z = X + Y
Divide both sides by 11:
Z = (X + Y) / 11

So, the solutions for X, Y, and Z in terms of each other are:

X = (X^2 - Y^2) / (3Y - Z - YZ)
Y = -(3Y - 3Z + X^2Z - ZX - 3XZ) / (XZ + XYZ - Z)
Z = (X + Y) / 11