The system of equations \(\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 2, \quad \frac{yz}{y + z} = 3\)
has exactly one solution. What is in this solution?
+23246 Flip each of the equations:
1) (xy) / (x + y) = 1 ---> (x + y) / (xy) = 1
2) (xz) / (x + z) = 2 ---> (x + z) / (xz) = 1/2
3) (yz) / (y + z) = 3 ---> (y + z) / (yz) = 1/3
Rewrite each equation:
1) (x + y) / (xy) = 1 ---> x / (xy) + y / (xy) = 1 ---> 1 / y + 1 / x = 1
2) (x + z) / (xz) = 1/2 ---> x / (xz) + z / (xz) = 1/2 ---> 1 / z + 1 / x = 1/2
3) (y + z) / (yz) = 1/3 ---> y / (yz) + z / (yz) = 1/3 ---> 1 / z + 1 / y = 1/3
Combining 1) and 2): 1 / y + 1 / x = 1
subtract: 1 / z + 1 / x = 1/2
1 / y - 1 / z = 1/2 (equation 4)
Combining 4) and 3): 1 / y - 1 / z = 1/2
add: 1 / z + 1 / y = 1/3
2 / y = 5/6
solving: y = 12/5
Now, substitute this back into equation 1) to find x and into equation 3) to find z.
