A dietitian is planning a meal containing 14 units of iron, 12 units of carbohy-

drates and 50 units of protein. Five ingredients are available. One portion of

each ingredient contains units of iron, carbohydrates and protein, as given in

the following table

I have attached, an image of the table

Suppose xi portions of ingredient number i are used, for i = 1; 2; 3; 4; 5. Then

three linear equations in $\displaystyle x_{1}; x_{2}; x_{3}; x_{4}; x_{5} $ must be satised. For example, the

iron requirement gives $\displaystyle x_{1} + 3x_{2} + 6x_{3} + 5x_{4} + 4x_{5} = 14. $

(a) Write down the augmented matrix of this system of three equations and

nd its reduced row-echelon form. Hence show that the solution can be

expressed in terms of arbitrary parameters s and t as

(x1; x2; x3; x4; x5) = (2; 4; 4; 0; 0) + s(1; 7; 7; 4; 0) + t(1; 19; 9; 0; 1).

(b) The amount of any ingredient used cannot be less than 0. Use this fact

to write down ve inequalities involving s and t. Show that t = 0 and

deduce that there is only one possible value of s. How many portions of

each ingredient should be used? (Fractions of a portion are allowed.)

I have worked out the reduce row echelon form for the equations I got

1 0 0 -1/4 1 |2

0 1 0 -7/4 19|-4

0 0 1 7/4 -9 |4

so the equations now are

$\displaystyle x_{1}-\frac{1}{4}x_{4} + x_{5} = 2 $

$\displaystyle x_{2} - \frac{7}{4} + 19_{5} = -4 $

$\displaystyle x_{3} + \frac{7}{4}x_{5} = 4 $

However I dont know how to get the solutions in the form of the parameter s and t? I am also stuck on part b,

any help appreciated