Answer:
Solving this system of equations, we find that the coordinates of the vertex are (3, 4).
Therefore, the coordinates of the vertex formed by the system of inequalities x ≤ 3, -x + 3y ≤ 12, and 4x + 3y ≥ 12 are (3, 4).
Explanation:
To find the coordinates of the vertices formed by the system of inequalities, we need to graph the equations and identify the points where the lines intersect.
1. Graph the equation x ≤ 3:
- Draw a vertical line at x = 3 on the x-axis.
2. Graph the equation -x + 3y ≤ 12:
- Rewrite the equation in slope-intercept form: 3y ≥ x + 12.
- Plot the line with a slope of 1/3 passing through the point (0, 4).
3. Graph the equation 4x + 3y ≥ 12:
- Rewrite the equation in slope-intercept form: 3y ≥ -4x + 12.
- Plot the line with a slope of -4/3 passing through the point (0, 4).
4. Shade the region that satisfies all the inequalities:
- Since x ≤ 3, shade the region to the left of the vertical line.
- Since -x + 3y ≤ 12, shade the region below the line.
- Since 4x + 3y ≥ 12, shade the region above the line.
5. Identify the vertices:
- The vertices are the points where the shaded regions intersect.
- In this case, there is a single vertex at the point of intersection between the lines -x + 3y ≤ 12 and 4x + 3y ≥ 12.
To find the coordinates of the vertex, we can solve the system of equations formed by the two lines:
-x + 3y = 12
4x + 3y = 12