The correct answer is d.
The graph shows a shaded region that is bounded by two lines. The first line has a slope of -1 and a y-intercept of 4. The second line has a slope of 3 and a y-intercept of 6.
To find the system of equations that represents this graph, we can start by writing down the equations for each of the two lines. The equation for the first line is:
y = -x + 4
The equation for the second line is:
y = 3x + 6
Now, we need to figure out which of the answer choices represents the region of the graph that is shaded. We can do this by looking at the signs of the inequalities in each answer choice.
Answer choice (a) has the following inequalities:
x ≥ 2
y + x ≤ 5
The first inequality means that all of the points in the shaded region must have an x-coordinate that is greater than or equal to 2. The second inequality means that all of the points in the shaded region must have a y-coordinate that is less than or equal to the sum of their x-coordinate and 5.
Answer choice (b) has the following inequalities:
y ≥ 2x + 1
y ≤ -x + 1
The first inequality means that all of the points in the shaded region must have a y-coordinate that is greater than or equal to twice their x-coordinate plus 1. The second inequality means that all of the points in the shaded region must have a y-coordinate that is less than or equal to the negative of their x-coordinate plus 1.
Answer choice (c) has the following inequalities:
y - x < 1
y - x > 3
The first inequality means that all of the points in the shaded region must have a y-coordinate that is less than their x-coordinate plus 1. The second inequality means that all of the points in the shaded region must have a y-coordinate that is greater than their x-coordinate plus 3.
Answer choice (d) has the following inequalities:
2x + y ≤ 4
3x - y ≥ 6
The first inequality means that all of the points in the shaded region must have a y-coordinate that is less than or equal to twice their x-coordinate plus 4. The second inequality means that all of the points in the shaded region must have a y-coordinate that is greater than or equal to three times their x-coordinate minus 6.
Answer choice (e) is empty, so it cannot be the correct answer.
By looking at the signs of the inequalities in each answer choice, we can see that answer choice (d) is the only answer choice that represents the region of the graph that is shaded. Therefore, the correct answer is **d.**
Here is a graph of the system of equations that represents the shaded region:
import matplotlib.pyplot as plt
def f(x):
return -x + 4
def g(x):
return 3x + 6
x = range(-6, 7)
y1 = [f(i) for i in x]
y2 = [g(i) for i in x]
plt.plot(x, y1, label='y = -x + 4')
plt.plot(x, y2, label='y = 3x + 6')
plt.fill_between(x, y2, y1, color='gray')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.grid(True)
plt.show()