The feasible region shown below is bounded by lines 2𝑥 − 𝑦 = 2 , 𝑥 + 𝑦 = 2 , and y = 0.
Find the coordinates of corner point A
2x - y = 4
x + y = 3
2x - y = 4
y = 0
x + y = 3
y = 0
.........................
3x = 7
x = 7/3
7/3 + y = 3
y = 3 - 7/3
y = 2/3
(7/3, 2/3)
2x = 4
x = 2
(2, 0)
x = 3
(3, 0)
corner points are
(7/3, 2/3), (2, 0), (3, 0)
To find the coordinates of corner point A, we need to identify the intersection points of the three lines that bound the feasible region.
First, let's find the intersection of lines 2𝑥 − 𝑦 = 2 and 𝑥 + 𝑦 = 2. To do this, we can solve these two equations simultaneously:
1. 2𝑥 − 𝑦 = 2
2. 𝑥 + 𝑦 = 2
To eliminate 𝑦, we can add equation 1 to equation 2:
3. 2𝑥 − 𝑦 + 𝑥 + 𝑦 = 2 + 2
Simplifying equation 3:
4. 3𝑥 = 4
Divide both sides of equation 4 by 3:
5. 𝑥 = 4/3
Substitute the value of 𝑥 into equation 2 to find 𝑦:
6. (4/3) + 𝑦 = 2
Subtract 4/3 from both sides of equation 6:
7. 𝑦 = 2 - 4/3
Further simplifying equation 7:
8. 𝑦 = 6/3 - 4/3 = 2/3
So the intersection point of lines 2𝑥 − 𝑦 = 2 and 𝑥 + 𝑦 = 2 is (4/3, 2/3).
Next, let's find the intersection of line 𝑥 + 𝑦 = 2 and y = 0. Since y = 0, we can substitute this value into equation 𝑥 + 𝑦 = 2:
9. 𝑥 + 0 = 2
Simplifying equation 9:
10. 𝑥 = 2
So the intersection point of line 𝑥 + 𝑦 = 2 and y = 0 is (2, 0).
Finally, there is a third intersection point between lines 2𝑥 − 𝑦 = 2 and y = 0 (since the feasible region has three vertices), but in this case, the y-coordinate is already given as y = 0. Thus, the third intersection point is (0, 0).
Therefore, the coordinates of corner point A (the intersection between the two lines) are (4/3, 2/3).