Solutions to Problem Set 1

1
.
​
1
.
1
.
,,,
1
.
2
.
,
1
.
3
.
,
1
.
4
.

2
.
​
2
.
1
.
Constant, Linear, Affine, Polynomial
2
.
2
.
Linear, Affine, Polynomial
2
.
3
.
Affine, Polynomial
2
.
4
.
Polynomial
2
.
5
.
Exponential
2
.
6
.
Constant, Linear, Affine, Polynomial (Note that that is just
log
1
2
which is a constant
3
.
​
3
.
1
.
No solutions:
x=y-1
according to the first equation, but when plugging into the second we get
3y-3=3y+6
which has no solution.
3
.
2
.
As above
x=y-1
, and plugging into the second equation we get
3y-3=y+3
. Simplifying we get
2y=6
so
y=3
and thus
x=2
4
.
​
4
.
1
.
2
(2x+3)
4
.
2
.
2
(2x+5)
4
.
3
.
2
(2x-5)
4
.
4
.
9(
3
x
-27)=9(x-3)(
2
x
+3x+9)
5
.
​
5
.
1
.
4
5
.
2
.
3-5=-2
5
.
3
.
-
log
2
(16)=-4
6
.
​
6
.
1
.
log
2
(x)+
log
2
(x)
log
2
(4)
=3
so ​
log
2
(x)1+
1
2
=3
so ​
log
2
(x)=2
so ​
x=4
6
.
2
.
2x
2
-
x
2
2=0
so
​
2
(
x
2
)
-
x
2
2=0
so
​
x
2
(
x
2
-2)=0,hence
​
​
x
2
=0
or
x
2
-2=0
thus
​
x=1
6
.
3
.
2x+2
10
=
x+1
50
so
​
2x
5
2x
2
100=
x
5
x
5
x
2
50
so
​
x
2
2=1
so
​
x=-1