1. ## linear equations

Hi there, Ive recently started on linear equations, and ive got this as an axample in the back of a text book i bought, but im not used to working with larger, written examples like this, if anyone could go through the motions with me, that'd be great....Thanks alot.

A box of Golden Oldies requires 0.3kg of wheat and 0.1kg of vitamins, and a box of Crunchy Bites requires 0.2kg of wheat, 0.1kg of sugar and 0.1kg of vitamins. Suppliers deliver at most 1400kg of wheat, at most 400kg of sugar and at least 500kg of the vitamins a week. The profit is £0.90 on a box of Golden Oldies and £1.00 on a box of Crunchy Bites.
Let X and Y and be the number of boxes of Golden Oldies and Crunchy Bites the manufacturer can produce per week.
(a) Show the manufacturer can make a weekly profit of P=0.9 X + Y

(b) Write down all the constraints for the problem.

(c) Solve the linear programming problem graphically to find the number of boxes of each type of cereal that maximise the profit. On your graph indicate clearly the feasible region, the optimal point, an arbitrary isoprofit line, and the isoprofit line corresponding to optimal profit.

2. Hello, bobchiba!

I'll give you the set-up . . . part (b).

A box of Golden Oldies requires 0.3kg of wheat and 0.1kg of vitamins.
A box of Crunchy Bites requires 0.2kg of wheat, 0.1kg of sugar and 0.1kg of vitamins.
Suppliers deliver at most 1400kg of wheat, at most 400kg of sugar
and at most 500kg of the vitamins a week.

The profit is £0.90 on a box of Golden Oldies and £1.00 on a box of Crunchy Bites.
Let $\displaystyle x$ and $\displaystyle y$ and be the number of boxes of Golden Oldies
and Crunchy Bites the manufacturer can produce per week.

(b) Write down all the constraints for the problem.

Organize the given information . . .

$\displaystyle \begin{array}{cccccccc}& | & \text{wheat} & | & \text{sugar} & | & \text{vitamins} & | \\ \hline \text{Golden Oldies }(x) & | & 0.3x & | & & | & 0.1x & | \\ \text{Crunchy Bites }(y) & | & 0.2y &|& 0.1y &|& 0.1y &| \\ \hline \text{available}& | & 1400 &|& 400 &|& 500 &|\end{array}$

The constraints are: .$\displaystyle \begin{Bmatrix}0.3x + 0.2y & \leq & 1400 \\ 0.1y & \leq & 400 \\ 0.1x + 0.1y & \leq & 500 \end{Bmatrix}$. . and, of course: .$\displaystyle \begin{Bmatrix}x & \geq & 0 \\ y & \geq & 0 \end{Bmatrix}$