119, Test1, Key
1.[14 points]
A candy manufacturer sells a snack mix for $1.5 a pound. The mix has three components: dry fruit
selling at $1 a pound; chocolate chips selling at $2 a pound; and nuts selling at $2 a pound. The mix
contains 2 pounds more of the chocolate chips than nuts. How many pounds of each component must be
used to make 40 pounds of the mixture?
a). Set up the system of linear equations to solve the above problem.
x+y+z=40
x+2y+2z=60
y-z=2
b). Set up the augmented matrix of the system and use row operations to solve the system
(you have to show all row operation steps for credit). Clearly state the solution set.
1 1 1

1 2 2
0 1  1
40

60 , final matrix is:
2 
1 0 0

0 1 0
0 0 1
20

10 
9 
Answer: x=20 pounds(Fruit), y=11 pounds (chips), z=9 pounds (nuts)
2.[12 points]
Each matrix given below is the reduced row-echelon form for the augmented matrix representing a
system of equations in either x and y or in x, y, and z. Determine which of the following is true for parts
a-d:
 There is a unique solution, (in that case give the solution set).
 There are an infinite number of solutions, (in that case list two of them).
 There is no solution, (in that case state the reason why).
1 0  1
4


2
a) 0 1 2
0 0 0
0
infinite# of sol.
x-z=4, y+2z=2 ex1: z=0, x=4, y=2,
ex2: z=1, x=5, y=0
1 0
5


2
c) 0 1
0 0
0
unique solution x=5, y=2
1 0 0
0


1
b) 0 1 0
0 0 1
0
unique sol. x=0, y=1, z=0
1 0

d) 0 1
0 0
4

2
1 
no solution (0=1 is false)
3.[12points]
Find x and y so that the following matrix equation is true.
 1  1  1 x 
1 1 1 3

2
 1 1   y 1 
1 1  3 1




 

x  1  2 2 1 3
 1 y
 1  y  x  1  2 2  3 1

 
 

3  y x  1 1 3
 y  1 3  x   3 1

 

3-y=1
y=2
x+1=3
x=2
y+1=3
y=2
3-x=1
x=2
Sol: x=2, y=2 (Make sure to check all 4 equations, sometimes you may not have a solution)
4.[12 points]
a) Select the value of coefficient k in the following system of equations so that the system will have
infinite many solutions.
6 x  7 y  z  2

yz 3


2 y  kz  6

Sol: k=-2 (second and third equations will be the same)
b) Give an example of 2x2 system of equations that does not have a solution.
System does not have a solution if lines are parallel, for ex:
x+y =5
x+y =7
5. [12 points]
Consider matrices below:
In parts a - f either
 Perform indicated matrix operation if it is possible. or
 Give the reason why, if the indicated matrix operation can't be performed
a) CD
[ 11 13
8]
d) AB
b) D-1
not a square matrix
c) AI=A
e) A+C
f)
.25 1.25
.25 A  

3.75 2 
(2x2)(3x1), inside dim.
must be the same
dimensions have to
be the same
Use the following system of linear inequalities is for next two problems (6 and 7)
x  2 y  6
2 x  y  6


 x0
 y  0
6.[12 points]
Graph the system. Clearly mark the solution region.
7.[12 points]
Find all corner points of the region you found in problem 6
and find the maximum and minimum of P = 2x + 3y over that region.
x
0
3
0
2
y
0
0
3
2
z
0
6
9
10
Zmax=10 at (2,2), Zmin =0 at (0,0)
Finding corner point (2,2): x+2y=6 /*(-2)
|-2x-4y=-12
2x+y=6
+ | 2x+y=6
Other corner points are x and y intercepts of your lines
-3y=-6
y=2
x+4=6
x=2
8.[14 points]
Consider the following system of equations:
3 x  7 y  13

2 x  5 y  9
a) Write the above system of equations in the form: AX=B
3 7  x  13
2 5  y   9 

   
b) Find A-1 , the inverse of a matrix A of your system. Clearly show all row operations you used.
A|I =
3
2

I|A-1 =
1 0
0 1

7 1
5 0
0
1
5
 7
2
3 
c) Find the solution of your system, X = A-1B.
Solution:
 5  7 13 2  x 
 2 3  9   1    y 

     
x=2, y=1