how would you solve this x^2+y^2=24 and y=2x+3

x^2+y^2=24 and y=2x+3  

Substituting  for y in the first equation, we have

x^2 + (2x + 3)^2  = 24    simplify

x^2 + 4x^2 + 12x + 9 = 24

5x^2 + 12x  - 15  = 0     this won't factor..so using the onsite solver, we have

$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{15}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{111}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}\right)}{{\mathtt{5}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{111}}}}{\mathtt{\,-\,}}{\mathtt{6}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{3.307\: \!130\: \!750\: \!570\: \!547\: \!8}}\\
{\mathtt{x}} = {\mathtt{0.907\: \!130\: \!750\: \!570\: \!547\: \!8}}\\
\end{array} \right\}$$

Let's round these to -3.31 and .91

And using y = 2x+ 3,  when x = -3.31, y = -3.614 and when x =.91, y = 4.814

So the solutions are (-3.31, -3.614) and ( .91, 4.814)

This is just the intersection of a line and a circle...here's the graph....https://www.desmos.com/calculator/oz2dnf8ems