Answer
$(-5,0), (3,72), (-3,18)$
Work Step by Step
We are given the system:
$\begin{cases}
y=x^3+5x^2\\
-9x+y=45
\end{cases}$
We will use the substitution method. Substitute the expression of $y$ from Equation 1 in Equation 2 to eliminate $y$ and determine $x$:
$-9x+x^3+5x^2=45$
$x^3+5x^2-9x-45=0$
$x^2(x+5)-9(x+5)=0$
$(x+5)(x^2-9)=0$
$(x+5)(x-3)(x+3)=0$
$x+5=0\Rightarrow x_1=-5$
$x-3=0\Rightarrow x_2=3$
$x+3=0\Rightarrow x_3=-3$
Substitute each of the values of $x$ in Equation 2 to determine $y$:
$-9x+y=45$
$x_1=-5\Rightarrow -9(-5)+y=45\Rightarrow 45+y=45\Rightarrow y_1=0$
$x_2=3\Rightarrow -9(3)+y=45\Rightarrow -27+y=45\Rightarrow y_2=72$
$x_3=-3\Rightarrow -9(-3)+y=45\Rightarrow 27+y=45\Rightarrow y_3=18$
The system's solutions are:
$(-5,0), (3,72), (-3,18)$
