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)