Simultaneous equation $x^2 +y=10050$ and $y^2 + x= 2600$ solution

Question

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.

Answer 1

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 $x=100\,$.

Answer 2

$\text{Jim} = x$, $\text{Tim} = y$ $$\begin{cases} x^2 +y=10050\\ y^2 + x= 2600\end{cases}$$ 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 $y = 50$.