Answer
$(1,4)$ and $(4,1)$.
Work Step by Step
The given system is
$y=x^2-6x+9$ ...... (1)
and $y+x=5$ ..... (2)
Solve for $y$ in the second equation.
$\Rightarrow y+x=5$
$\Rightarrow y=5-x$ ......(3)
Substitute the value of $y$ into equation (1).
$\Rightarrow 5-x=x^2-6x+9$
Move all terms to the one side.
$\Rightarrow 0=x^2-6x+9-5+x$
Simplify.
$\Rightarrow 0=x^2-5x+4$
Factor.
$\Rightarrow 0=(x-1)(x-4)$
Use the zero product property.
$x-1=0$ or $x-4=0$
Solve for $x$.
$x=1$ or $x=4$
Substitute back the value of $x$ into equation (2) and solve for $y$.
For $x=1$.
$\Rightarrow y+1=5$
$\Rightarrow y=5-1$
$\Rightarrow y=4$
For $x=4$.
$\Rightarrow y+4=5$
$\Rightarrow y=5-4$
$\Rightarrow y=1$
The points are $(1,4)$ and $(4,1)$.
