Simultaneous equation $x^2 +y=10050$ and $y^2 + x= 2600$ solution Here's the problem:
Jim and Tim are sharing money. If I square Jim’s money and add on Tim’s, I get £10,050. If I square Tim’s money and add on Jim’s, I get £2,600. How much do they each have?
From this the equations are as follows  $x^2 +y=10050$ and $y^2 + x= 2600$
This is a grade 10 math problem and want to solve it w/o using differentiation, integration etc.
Hint: assuming whole numbers, $y^2 \le 2600 \implies y \le 50\,$ and $x^2 \le 10050 \implies x \le 100\,$. But then $x^2 = 10050-y \ge 10050-50 = 10000\,$ so $x=100\,$.
$\text{Jim} = x$, $\text{Tim} = y$
$$\begin{cases} x^2 +y=10050\\
y^2 + x= 2600\end{cases}$$
Subtracting the equations, 
$$x^2 – y^2 +y – x = 10050-2600=   7450
\\( x-y) (x+y-1)= 7450 = 149 \times 50$$
Since $149$ is prime and $x\ge y$,
$$\begin{cases} x- y = 50 \\
x + y -1 = 149\end{cases}$$
After solving $x= 100$ and $y = 50$.