Answer
$(1,-2)$ and $(2,1)$
Work Step by Step
$\left\{\begin{aligned}y&=3x-5\\x^{2}+y^{2}&=5\end{aligned}\right.$
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Substitute y into the second equation
$\begin{aligned}x^{2}+(3x-5)^{2}&=5\\x^{2}+9x^{2}-30x+25&=5\\10x^{2}-30x+20&=0\\x^{2}-3x+2&=0\end{aligned}$
... two factors of 2 whose sum is -3 ... are -1 and -2.
$(x-1)(x-2)=0$
Back-substituting
$\left[\begin{array}{lll}
x=1 & ... & x=2\\
y=3(1)-5 & & y=3(2)-5\\
y=-2 & & y=1\\
& & \\
(1,-2) & & (2,1)
\end{array}\right]$
