Answer:
The vertices of the triangle are (2.625, 5.125), (0.645, 5.52), and (5.818, 0.182).
Explanation:
To find the vertices of the triangle formed by the system of inequalities, we need to find the intersection points of each pair of lines.
1. First Inequality:
5y ≤ 2x + 9
Rewrite to express y in terms of x:
y ≤ (2/5)x + (9/5)
2. Second Inequality:
y ≤ -x + 6
3. Third Inequality:
9y ≥ -2x + 5
Divide both sides by 9:
y ≥ (-2/9)x + (5/9)
4. Intersections:
First and Second Inequality:
Set both inequalities equal to each other:
(2/5)x + (9/5) = -x + 6
Solve for x:
3x + 9 = -5x + 30
8x = 21
x = 2.625
Substitute x back into either inequality to find y:
y ≤ (-2/5) * 2.625 + 6
y ≤ 5.125
Intersection point 1: (2.625, 5.125)
First and Third Inequality:
Set both inequalities equal to each other:
(2/5)x + (9/5) = (-2/9)x + (5/9)
Solve for x:
21x + 45 = -10x + 25
31x = 20
x = 0.645
Substitute x back into either inequality to find y:
y ≤ (-2/5) * 0.645 + 6
y ≤ 5.52
Intersection point 2: (0.645, 5.52)
Second and Third Inequality:
Set both inequalities equal to each other:
-x + 6 = (-2/9)x + (5/9)
Solve for x:
9x - 54 = -2x + 10
11x = 64
x = 5.818
Substitute x back into either inequality to find y:
y ≤ -5.818 + 6
y ≤ 0.182
Intersection point 3: (5.818, 0.182)
Thus, the vertices of the triangle are (2.625, 5.125), (0.645, 5.52), and (5.818, 0.182).