# x+y=5 and x^y+y^x=17 Find the value of x and y?

Jun 19, 2017

$x = 2 , y = 3$
Assuming $\left\{x , y\right\} \in \mathbb{N}$

#### Explanation:

Here we have two equations:

$x + y = 5$

${x}^{y} + {y}^{x} = 17$

{Although there is no resaon to assume that $x$ and $y$ are naturals, it seemed like a sensible place to start.}

Assume $\left\{x , y\right\} \in \mathbb{N}$

Since $x + y = 5$

$x \in \left\{1 , 2 , 3 , 4\right\}$ and $y \in \left\{4 , 3 , 2 , 1\right\}$

Testing each $\left(x , y\right)$ pair in turn we notice that:

${2}^{3} + {3}^{2} = 8 + 9 = 17$

Hence a solution to this system is $x = 2 , y = 3$

NB: I have not worked through the cases where $\left(x , y\right) \in \mathbb{R}$, Hence, I have not proved that there are no other real solutions to this system.