On a movie set, an archway is modeled by the equation y = -0.5x2 + 3x, where y is the height in feet and x is the horizontal distance in feet. a laser is directed at the archway at an angle modeled by the equation -0.5x + 2.42y = 7.65 such that the beam crosses the archway at points a and b. at what height from the ground are the points a and b?

This question really just needs you to solve the system of equations.  These solutions will be the two points when the curve (archway) and the line (laser) intersect.

Let us express the laser line in terms of y.

-.5x+2.42y=7.65

2.42y=.5x+7.65

y=(.5x+7.65)/2.42

The curve is just y=-.5x^2+3x so when it intersects the curve, y=y and x=x so  we can say:

(.5x+7.65)/2.42=-.5x^2+3x

.5x+7.65=-1.21x^2+7.26x

1.21x^2-6.76x+7.65=0  using the quadratic for simplicity and solving for x:

x=(6.76±√8.6716)/2.42

x≈1.577, 4.010  (to nearest thousandth)

So the heights can found using the simple linear equation y=(.5x+7.65)/2.42 with those values of x...

y≈3.49, 3.99