math final
mometag
m181_sample_final_fall_15.docMATHEMATICAL ANALYSIS I, MATH 181
Final
1. Represent each pair of statements as a system of two equations. Verify the given values for x and y are a solution of the system of equations.
(a) The sum of two numbers if twenty-five and twice the first number added to the second number totals thirty-two. Let x be the first number and y be the second number. x = 7 and y = 18.
(b) An envelope of $10 and $20 bills contains eight bills and the money in the envelope is worth $110. Let x be the number of $10 bills and y be the number of $20 bills. x = 5 and y = 3.
2. Solve each equation by graphing. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.
(a)
(
)
î
í
ì
=
+
=
-
=
6
2
2
2
y
x
y
x
x
f
(c)(
)
î
í
ì
-
=
+
-
=
-
=
8
2
16
4
2
y
x
y
x
x
f
3. Solve each equation by the substitution method. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.
(a)
(
)
î
í
ì
=
=
+
=
2
8
2
x
y
x
x
f
(b)(
)
î
í
ì
=
-
=
+
=
5
25
3
2
y
x
y
x
x
f
(c)(
)
î
í
ì
=
+
-
=
-
=
30
4
6
30
2
3
y
x
y
x
x
f
(
)
î
í
ì
=
+
=
+
-
=
4
2
y
x
y
x
x
f
4. Solve each equation by the elimination method. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.(a)
(
)
î
í
ì
=
+
=
+
=
13
30
2
3
y
x
y
x
x
f
(b)(
)
î
í
ì
=
+
=
+
=
10
2
30
3
y
x
y
x
x
f
(c)(
)
î
í
ì
=
-
=
+
=
10
30
3
2
y
x
y
x
x
f
5. Find the augmented matrix representing the system of equations.
(a)
î
í
ì
=
+
-
=
-
60
4
3
10
2
y
x
y
x
(b)î
í
ì
=
=
-
8
24
3
4
y
y
x
(c)î
í
ì
=
=
12
18
x
y
6. Find the system of equations represented by the augmented matrix.
(a)
÷
÷
ø
ö
ç
ç
è
æ
-
5
3
1
1
0
1
(b)÷
÷
ø
ö
ç
ç
è
æ
7
5
3
2
1
1
(c)÷
÷
ø
ö
ç
ç
è
æ
-
-
48
60
12
16
15
20
7. Add or subtract the following matrices.
÷
÷
ø
ö
ç
ç
è
æ
-
-
+
÷
÷
ø
ö
ç
ç
è
æ
-
7
2
8
4
6
3
5
2
(b)÷
÷
ø
ö
ç
ç
è
æ
-
-
-
÷
÷
ø
ö
ç
ç
è
æ
-
5
3
1
2
8
9
6
4
8. Multiply the following matrices.
(
)
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
4
5
2
`
1
4
3
(b)÷
÷
ø
ö
ç
ç
è
æ
-
-
÷
÷
ø
ö
ç
ç
è
æ
-
7
9
3
1
3
5
4
2
9. Solve each system by row-reducing the corresponding augmented matrix. If the solution is not unique, identify the system as ‘inconsistent’ or ‘dependent’.
(a)
î
í
ì
=
=
+
6
8
y
y
x
(b)î
í
ì
=
+
=
-
3
6
2
y
x
y
x
(c)î
í
ì
=
+
=
+
4
2
9
3
y
x
y
x
10. For each matrix A, find the inverse matrix, A-1, or identify A as a singular matrix.
(a)
÷
÷
ø
ö
ç
ç
è
æ
1
0
4
1
(b)÷
÷
ø
ö
ç
ç
è
æ
3
2
7
5
� EMBED \* MERGEFORMAT ���
