Problem: The product of three numbers is 54. The sum of the numbers is 14. What are the three numbers
This is for my son and I feel stupid for not being able to figure this out.
1) x * y * z = 54
2) x + y + z = 14
We have only two equations, with three unknowns. We need to
choose a value for one of them and recalculate the equations.
This will give us two equations with two unknowns. Let's set
x equal to 2.
x * y * z = 54
2 * y * z = 54
3) yz = 27
x + y + z = 14
2 + y + z = 14
y + z = 12
4) y = 12 - z
Substitute the value of y from equation 4 into equation 3 and solve for z.
(12 - z)z = 27
12z - z^2 = 27
12z - z^2 - 27 = 27 - 27
-z^2 + 12z - 27 = 0
z^2 - 12z + 27 = 0
(z - 9)(z - 3) = 0
Set each factor to zero and solve for z.
z - 9 = 0
z = 9
z - 3 = 0
z = 3
We can see from equation 3, yz = 27, that whichever value
we choose for z, y will be the other one.
So we have x = 2, y = 3, z = 9
and x = 2, y = 9, z = 3.
We can, in fact, set x equal to either 3 or 9, and y and z
will change values as needed.
The important point is that the three numers are: 2, 3 and 9.