Final answer:
The area of the shaded region, which is a right triangle formed by the intersection of the lines y = -2x + 8 and y = (2/3)x + (16/3) at point (-2, 12), is 4 square units.
Step-by-step explanation:
To determine the area of the shaded region formed by the equations y = -2x + 8 and 3y = 2x + 16, we first need to find the points of intersection of the two lines. To do this, we set the two equations equal to each other and solve for x.
The second equation can be simplified to y = (2/3)x + (16/3). Setting the two equations equal, we have:
-2x + 8 = (2/3)x + (16/3)
Adding 2x to both sides and multiplying by 3 to clear fractions we get:
6x + 24 = 2x + 16
Subtracting 2x from both sides we get:
4x + 24 = 16.
Subtracting 24 from both sides, we have:
4x = -8
Dividing both sides by 4 gives us:
x = -2
Substituting x back into the first equation gives us:
y = -2(-2) + 8 = 4 + 8 = 12
Thus, the intersection point is (-2, 12).
Now that we have the intersection point, we can find the area of the shaded region, which is a right triangle with vertices (0, 8), (-2, 12), and (0, 12). The base of the triangle has a length of 2 units (from x=0 to x=-2), and the height is 4 units (from y=8 to y=12). The formula for the area of a right triangle is (1/2) * base * height, so the area of the shaded region is:
(1/2) * 2 * 4 = 4 square units.