linear programming programming by graphing and then determining which vertex minimizes the objective function G=4x+3y .
{5x+15y≥15
{5x+5y≥35
{x≥0
{y≥0
x =
y =
What is the minimum value? G=
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To solve this linear programming problem using graphing, we need to start by graphing the feasible region defined by the given inequalities.
1. Start by graphing the first inequality: 5x + 15y ≥ 15.
- Rearrange the inequality to slope-intercept form: y ≥ (15 - 5x) / 15
- Set x = 0 and find the corresponding y-coordinate. Plot the point (0, 1).
- Set y = 0 and find the corresponding x-coordinate. Plot the point (3, 0).
- Connect these two points to form a line.
2. Next, graph the second inequality: 5x + 5y ≥ 35.
- Rearrange the inequality to slope-intercept form: y ≥ (35 - 5x) / 5
- Set x = 0 and find the corresponding y-coordinate. Plot the point (0, 7).
- Set y = 0 and find the corresponding x-coordinate. Plot the point (7, 0).
- Connect these two points to form a line.
3. Finally, graph the non-negativity constraints: x ≥ 0 and y ≥ 0.
- Draw vertical and horizontal lines passing through the origin (0,0).
Now, shade the feasible region, which is the intersection of the shaded regions formed by the two inequality lines and the quadrant defined by the non-negativity constraints.
After graphing the feasible region, we need to find the vertex (corner point) that minimizes the objective function G = 4x + 3y. To do this, we evaluate the objective function at each vertex:
1. Identify the vertices of the feasible region by finding the points where the lines intersect.
- The points where the two lines intersect are the vertices of the feasible region.
2. Plug the x and y values of each vertex into the objective function G = 4x + 3y to find the corresponding minimum values.
3. Compare the objective function values at each vertex to determine the minimum value.
After calculating the objective function at each vertex, the minimum value of G can be determined by comparing the results. The vertex that corresponds to the smallest value of G will be the minimum value.