Answer
The solutions are $(2,-9)$ and $(8,9)$.
Work Step by Step
The given system is
$y=3x-15$ ...... (1)
$y=\frac{1}{2}x^2-2x-7$ ...... (2)
Graph each equation.
The point of intersections are $A=(2,-9)$ and $B=(8,9)$.
Check: $(x,y)=(2,-9)$
Equation (1).
$\Rightarrow y=2x-5$
$\Rightarrow -9=3(2)-15$
$\Rightarrow -9=6-15$
$\Rightarrow -9=-9$
True.
Equation (2).
$\Rightarrow y=\frac{1}{2}x^2-2x-7$
$\Rightarrow -9=\frac{1}{2}(2)^2-2(2)-7$
$\Rightarrow -9=2-4-7$
$\Rightarrow -9=2-11$
$\Rightarrow -9=-9$
True.
Check: $(x,y)=(8,9)$
Equation (1).
$\Rightarrow y=2x-5$
$\Rightarrow 9=3(8)-15$
$\Rightarrow 9=24-15$
$\Rightarrow 9=9$
True.
Equation (2).
$\Rightarrow y=\frac{1}{2}x^2-2x-7$
$\Rightarrow 9=\frac{1}{2}(8)^2-2(8)-7$
$\Rightarrow 9=32-16-7$
$\Rightarrow 9=32-23$
$\Rightarrow 9=9$
True.
Hence, the solutions are $(2,-9)$ and $(8,9)$.
