Solve the 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 a linear programming problem by graphing, we need to plot the inequalities on a graph and then identify the feasible region. The objective is to find the vertex of the feasible region that minimizes the objective function.
Let's start by graphing the given inequalities:
1) 5x + 15y ≥ 15
2) 5x + 5y ≥ 35
3) x ≥ 0
4) y ≥ 0
To graph the first inequality, we can rewrite it as an equation: 5x + 15y = 15.
We can find two points on this line by setting x to zero and then y to zero:
At x = 0, we have 15y = 15, so y = 1. Giving us the point (0,1).
At y = 0, we have 5x = 15, so x = 3. Giving us the point (3,0).
Plot these two points and draw a line passing through them, extending it infinitely in both directions. Note that since the inequality is greater than or equal to, the line should be solid (not dashed).
Next, let's graph the second inequality: 5x + 5y ≥ 35.
We can again rewrite this inequality as an equation: 5x + 5y = 35.
Setting x to zero, we get 5y = 35, so y = 7. Giving us the point (0,7).
Setting y to zero, we get 5x = 35, so x = 7. Giving us the point (7,0).
Plot these two points and draw a line passing through them indefinitely. Again, make sure the line is solid.
Next, let's graph the third inequality: x ≥ 0.
This inequality represents the x-axis. Simply draw a vertical line passing through the y-axis at x = 0. The line should be solid since x is greater than or equal to zero.
Lastly, let's graph the fourth inequality: y ≥ 0.
This inequality represents the y-axis. Draw a horizontal line passing through the x-axis at y = 0. The line should also be solid since y is greater than or equal to zero.
Now, look at the region where all the shaded areas overlap. This region is called the feasible region.
To optimize the objective function G = 4x + 3y, we need to evaluate this function at each vertex of the feasible region (corners of the shaded area) and find the smallest value.
In this case, I cannot determine the exact vertices without the graph or the graph coordinates. However, once you have the graph, you can easily identify the vertices by finding the intersection points of the lines.
Once you have the vertices, substitute the x and y values of each vertex into the objective function G = 4x + 3y and calculate the corresponding value. The vertex with the smallest value of G will be the solution.
To find the minimum value of G, you need to evaluate G at each vertex of the feasible region and choose the vertex with the smallest G value.