Word Puzzle in Three Variables

Question

This is a puzzle that I translated into equations. It goes like this:

There are a total of 20 people in a room (male, female and children). Men eat 2 pieces of bread per person, women eat 1.5 pieces of bread and children eat 0.5. There are a total of 20 pieces of bread. What is the number of men women and children?

So we have $2x + 1.5y + 0.5z = 20$ and $x + y + z = 20$. None of the variables can be zero. How to solve?

Answer 1

After multiplying the first equation by two to get $4x+3y+z=40$, we have two Diophantine equations in three variables. Subtracting $x+y+z=20$ from $4x+3y+z=40$ we get $3x+2y=20$, which has the following solutions with $x,y\in\Bbb N$: $$(x,y)=(2,7),(4,4),(6,1)$$ These translate into three solutions for the numbers of different types of people in the room: $$(x,y,z)=(2,7,11),(4,4,12),(6,1,13)$$

Answer 2

3x+2Y=20

1. X must be less than 7, because 3x=21 which is greater than 2o and y cannot be negative.

2. X is limited to values 0,1,2,3,4,5,6 try each one

3. X can be 2, 4, 6 Three solutions

Note: x must be even.