Page 78 - Matrix Analysis & Applied Linear Algebra

2.5 Nonhomogeneous Systems 71

2.5.2. Among the solutions that satisfy the set of linear equations

x 1 + x 2 +2x 3 +2x 4 + x 5 =1,
2x 1 +2x 2 +4x 3 +4x 4 +3x 5 =1,
2x 1 +2x 2 +4x 3 +4x 4 +2x 5 =2,
3x 1 +5x 2 +8x 3 +6x 4 +5x 5 =3,

find all those that also satisfy the following two constraints:
2
2
(x 1 − x 2 ) − 4x =0,
5
2
2
x − x =0.
3 5
2.5.3. In order to grow a certain crop, it is recommended that each square foot
of ground be treated with 10 units of phosphorous, 9 units of potassium,
and 19 units of nitrogen. Suppose that there are three brands of fertilizer
on the market—say brand X, brand Y, and brand Z. One pound of
brand X contains 2 units of phosphorous, 3 units of potassium, and 5
units of nitrogen. One pound of brand Y contains 1 unit of phosphorous,
3 units of potassium, and 4 units of nitrogen. One pound of brand Z
contains only 1 unit of phosphorous and 1 unit of nitrogen.
(a) Take into account the obvious fact that a negative number of
pounds of any brand can never be applied, and suppose that
because of the way fertilizer is sold only an integral number of
pounds of each brand will be applied. Under these constraints,
determine all possible combinations of the three brands that can
be applied to satisfy the recommendations exactly.
(b) Suppose that brand X costs $1 per pound, brand Y costs $6
per pound, and brand Z costs $3 per pound. Determine the
least expensive solution that will satisfy the recommendations
exactly as well as the constraints of part (a).

2.5.4. Consider the following system:

2x +2y +3z =0,
4x +8y +12z = −4,
6x +2y + αz =4.
(a) Determine all values of α for which the system is consistent.
(b) Determine all values of α for which there is a unique solution,
and compute the solution for these cases.
(c) Determine all values of α for which there are infinitely many
different solutions, and give the general solution for these cases.