Solve the system y = -x + 7 and y = -0.5(x - 3)^2 + 8

Both equal y so set equal to each other
-x+7=-0.5(x-3)^2+8
times both sides by -2
2x-14=(x-3)^2-16
expand
2x-14=x^2-6x+9-16
add 14-2x to both sides
0=x^2-8x+7
factor
0=(x-7)(x-1)
set to zero
0=x-7
7=x

0=x-1
1=x

so find y

y=-x+7
y=-1+7
y=6
or

y=-7+7
y=0

so the solutions are
(-1,6) and (-7,0)