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 must be satised. For example, the

iron requirement gives

(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

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