Sum of integers and system of equations

Question

I always wondered but never understood how to solve the following type of equation $$\begin{matrix} &{x}+y=35 \\ &x\geq15 \\ &y\geq 15 \\ &(x,y) \in \mathbb{N} \end{matrix} $$

(I just made up values) I found the two pairs $$(19;16) \;\& \;(17;18)$$ Are they the only solutions ? if they are, how to prove it?

Answer 1

You can write $ x=15+k_1, y =15+k_2$ Now substitute these in $ x+y =35 $ We get $ k_1 + k_2 =5 $ Now the number of possible values of $k_1, k_2$ corresponds to the possible values of $x$ and $y$ so the number of ways are $ C(n+r-1,r-1) $ Here, in this particular problem $n=5$ and $r=2$, so total ways are $6$.

Answer 2

Given that $\ x\ge 15$, we get that, $\ 35-y \ge 15$ i.e $\ y\le 20$. Since , its given $\ y\ge 15$ . Therefore , possible integral values of $\ y$ belong to the set $ \{\,15,16,17,18,19,20\,\}$

And for each $y$ , there is a unique value of $x=35-y$ that is an integer.

Answer 3

Graph!

And also, there are infinitely many rational solutions, however only a finite amount of integer coordinates.

Here is the graph:

The intersection of the green and blue shaded sections (green-blue section) give you your $x$ and $y$ solutions.