<s>[INST] Solve equation with two unknowns I have these equations.
$2\pi r_1+2\pi r_2=24$
and $\pi r_1^2+\pi r_2^2=20$
and to solve them I did the following steps
Step 1 : $\frac {2\pi r_1+2\pi r_2}{2}= \frac {24}{2}$
Step 2 : $\pi r_1+\pi r_2= 12$
Step 3 : $\pi r_1= 12 - \pi r_2$
Step 4 : $\pi r_1^2+(12 - \pi r_2)^2=20$
Step 5 : $\pi r_1^2+(12 - \pi r_2)^2=20$
Step 6 : $\pi r_1^2+144 - 24\pi r_2 + \pi r_1^2=20$
Step 7 : $2\pi r_1^2- 24\pi r_2+124=0$
Step 8 : $\pi r_1 = - \frac {-24}{2} \pm \sqrt{(\frac {-24}{2})^2 - 124}$
I get the result $6 \pm \sqrt{10}$ but is this correct? Something feels wrong about this answer but don't know what.
Thanks!
 [/INST] $\begin{cases}2\pi r_1+2\pi r_2=24 & (1) \\ \pi r_1^2+\pi r_2^2=20 & (2)\end{cases}$
Equation $(1)$ gives $r_1=\frac{12-\pi r_2}{\pi}$
Plug that into $(2)$, we have $\pi\big(\frac{12-\pi r_2}{\pi}\big)^2+\pi r_2^2=20 \Rightarrow 2\pi r_2^2-24r_2+\frac{144}{\pi}-20=0$.
This quadratic equation has no solutions, and therefore there is no solution to the system of equations.

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