I keep trying to set up this problem to solve it using the Simplex Method, but I can't seem to get the tableau set up correctly. If anyone could help, I would appreciate it:

A baker has 150 units of flour, 90 of sugar, and 150 of raisins. A loaf of raisin bread requires 1 unit of flour, 1 of sugar and 2 of raisins, while a raisin cake needs 5, 2 and 1 units respectively.

If raisin bread sells for $1.75 a loaf and raisin cake for $4.00 each, how many of each should be baked so that gross income is maximized?

Can you show us what you've got?

To solve this problem using the Simplex Method, we need to set up a linear programming model. Let's define our decision variables and constraints first.

Let x be the number of loaves of raisin bread and y be the number of raisin cakes to be baked.

Objective function:
We want to maximize the gross income. The gross income for each loaf of raisin bread is $1.75, and for each raisin cake is $4.00. Therefore, the objective function is:
Z = 1.75x + 4.00y

Constraints:
1. The baker has a limited amount of flour: 1x + 5y ≤ 150 (flour constraint)
2. The baker has a limited amount of sugar: 1x + 2y ≤ 90 (sugar constraint)
3. The baker has a limited amount of raisins: 2x + y ≤ 150 (raisin constraint)
4. The number of loaves of raisin bread and raisin cakes should be greater than or equal to zero: x ≥ 0, y ≥ 0

Now, let's convert these inequalities into equations by introducing slack variables to get a standard form of linear programming.

1. x + 5y + s1 = 150
2. x + 2y + s2 = 90
3. 2x + y + s3 = 150
4. x, y, s1, s2, s3 ≥ 0

To set up the initial tableau for the Simplex Method, we create the following table:

| x | y | s1 | s2 | s3 | RHS |
----------------------------------------------
s1 | 1 | 5 | 1 | 0 | 0 | 150 |
s2 | 1 | 2 | 0 | 1 | 0 | 90 |
s3 | 2 | 1 | 0 | 0 | 1 | 150 |
----------------------------------------------
Z | -1.75| -4 | 0 | 0 | 0 | 0 |

Here, x, y, s1, s2, s3 represent the decision variables and slack variables on the left side of the table. The right-hand side (RHS) represents the right side of the constraints, and Z represents the coefficients of the objective function.

At this point, we can use the Simplex Method to optimize the solution by finding the pivot element and performing row operations. However, solving the entire linear programming problem using the Simplex Method requires multiple iterations, pivoting, and row operations, which might be quite lengthy to explain.

It is recommended to use software programs or online tools that can perform the Simplex Method for you automatically. There are many linear programming solvers available online that can quickly solve these types of optimization problems. You can input the objective function and constraints, and the solver will provide the optimal solution for you.

I hope this explanation helps you set up the problem using the Simplex Method. Let me know if you have any further questions!