Answer
$(-0.41,-10.71)$, $(2.41,4.07)$
Work Step by Step
Given $$
\begin{cases}
y&=x^2+3 x-9 \\
y&=5 x-8.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$.
$$
\begin{aligned}
x^2+3 x-9&=5 x-8 \\
x^2+3 x-5 x-9+8&=0 \\
x^2-2 x-1&=0 \\
x^2-2 x+\left(\frac{2}{2}\right)^2&=1+\left(\frac{2}{2}\right)^2\\
x^2-2 x+1^2&=2 \\
(x-1)^2&=2 \\
x-1&= \pm \sqrt{2} \\
x&=1 \pm \sqrt{2}
\end{aligned}
$$ We have: $$
\begin{aligned}
x & =1-\sqrt{2} \\
& =-0.414214 \\
x & =1+\sqrt{2} \\
& =2.414214
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations. $$
\begin{aligned}
y&=5 (1-\sqrt 2)-8\\
& = -3-5\sqrt 2\approx -10.71\\
y&=5 (1+\sqrt 2)-8\\
& = -3+5\sqrt 2\approx 4.071.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-0.41,-10.71)$ and $(2.41,4.07)$.
See the graph.
