Math 484.1  September 29, 2011  Name:_________Dr.V.________________________________

Midterm 1,  5 problems, 15 points each.  Return this page with your name on both sides.

1. Solve for x and y where a is a given number:

a2x + y= a2,

ax +ay = 1.

Solution. A row addition operation gives

a2x + y= a2,

(a - a3 )x = 1 - a3.

If a  ≠ 0 , ±1, then  x = (1 - a3 )/(a - a3 )=  (1+ a +a2 )/(a + a2 ) and

y = (1-x)a2 =  -a/(1+a).

If  a = 0 or -1, then  there are no solutions.

If a = 1, then y = 1 - x (x arbitrary).

2. 4x + y 2 -> min,

x2 +  y 2 = 10; x and y integers.

Solution There are 8 feasible solutions: (x, y) = (±1, ±3), (±3, ±1).

min =- -11 at  x = -3, y = ±1 (two optimal solutions).

3, 4. Solve the linear programs given by the following tableaux with all decision variables xi ≥ 0:

 x1 x2 x3 -1 Problem 3 -1 0 -1 2 = x4 1 0 -1 -1 -> min

Solution. The standard tableau is

 x1 x2 x3 1 Problem 3 -1 0 -1 -2 = x4 1 0 -1 1 -> min

The  x4-row is bad, so LP is feasible.

 x1 x2 -x3 1 Problem 4 1 0 1 -2 =- x4 1 0 -1 -1 -> min

Solution. . The standard tableau is

 x1 x2 x3 1 Problem 4 -1 0 1 2 = x4 1 0 1 -1 -> min

It is optimal so  min = -1 at   x1  =x2  =  x3  = 0,  x4 = 2.

5. Find all logical implications between the following 5 constraints on  x, y:

(a)   x2 = y2, (b) 0 <   -2, (c) 0 < 1, (d) x = -y, (e) x=y=0.

Solution.

(b)   (e)   (d)     (a)    (c) .