To find the area of the triangle formed by the given equations, we need to find the vertices of the triangle and then use the formula for the area of a triangle.
Given equations:
1) x = 0
2) 2x - 3y = 6
3) x + y = 3
To find the vertices, we can solve the equations to find the points of intersection.
Solving equations 2) and 3) simultaneously:
2x - 3y = 6
x + y = 3
Multiplying the second equation by 2 to eliminate x:
2(x + y) = 2(3)
2x + 2y = 6
Now we have the system of equations:
2x - 3y = 6
2x + 2y = 6
Subtracting the equations to eliminate x:
(2x - 3y) - (2x + 2y) = 6 - 6
-5y = 0
y = 0
Substituting y = 0 into equation 2):
2x - 3(0) = 6
2x = 6
x = 3
The point of intersection of equations 2) and 3) is (3, 0).
Next, we need to find the point of intersection of equations 1) and 2).
Substituting x = 0 into equation 2):
2(0) - 3y = 6
-3y = 6
y = -2
The point of intersection of equations 1) and 2) is (0, -2).
Lastly, we need to find the point of intersection of equations 1) and 3).
Substituting x = 0 into equation 3):
0 + y = 3
y = 3
The point of intersection of equations 1) and 3) is (0, 3).
Now we have the three vertices of the triangle: (3, 0), (0, -2), and (0, 3).
To find the area of the triangle, we can use the formula for the area of a triangle given its vertices:
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Substituting the coordinates of the vertices into the formula:
Area = 1/2 * |3(3 - (-2)) + 0((-2) - 0) + 0(0 - 3)|
Area = 1/2 * |3(5) + 0 + 0|
Area = 1/2 * |15|
Area = 1/2 * 15
Area = 7.5
Therefore, the area of the triangle formed by the given equations is 7.5 square units, which is equivalent to option B).