Given:
[tex]\begin{gathered} x^2+y^2=25\ldots\ldots(1) \\ x+y=7\ldots\ldots\text{.}(2) \end{gathered}[/tex]To solve for x and y:
Using the formula,
[tex](x+y)^2=x^2+y^2+2xy[/tex]Substitute the given values we get,
[tex]\begin{gathered} 7^2=25+2xy \\ 2xy=49-25 \\ 2xy=24 \\ xy=12 \\ y=\frac{12}{x}\ldots\ldots\ldots(3) \end{gathered}[/tex]Substitute equ (3) in (2), we get,
[tex]\begin{gathered} x+\frac{12}{x}=7 \\ \frac{x^2+12}{x}=7 \\ x^2+12=7x \\ x^2-7x+12=0 \\ (x-4)(x-3)=0 \\ x=4(or)3 \end{gathered}[/tex]Substitute x=4 and x=3 in equation (3) we get,
[tex]\begin{gathered} y=\frac{12}{4} \\ y=3 \\ y=\frac{12}{3} \\ y=4 \end{gathered}[/tex]Hence, the solutions are,
[tex](4,\text{ 3) and (3,4)}[/tex]