No solution!  Here's why:

Let the number of nickels be x
Then the number of dimes, using 
ONE PART = TOTAL MINUS OTHER PART,
is 22-x.  
                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
NICKELS      x        $0.05    $0.05x 
DIMES      22-x       $0.10    $0.10(22-x)
-------------------------------------------
TOTALS      22        -----    $2.35

 The equation comes from the column on the right

  
0.05x + 0.10(22-x) = 2.35
Get rid of decimals by multiplying every term by 100
     5x + 10(22-x) = 235
    5x + 220 - 10x = 235
         -5x + 220 = 235
               -5x = 15
                 x = -3 = the number of nickels
The number of dimes is 22-x or 22-(-3) or 25 dimes.

Checking:  -3 nickels is -$0.15 and 25 dimes is $2.50

And -$0.15+2.50 = $2.35.  

However, the number of nickels cannot be negative, so

THERE IS NO SOLUTION.

Further reason for "no solution.  Here are all the ways you
can have 22 coins in nickels and dimes.  You can see that
none of them add up to $2.35. The most you can have is $2.15
and the least you can have is $1.15.  So either you copied
the problem wrong o else the answer is "no solution".

1 nickel and 21 dimes is $2.15.
2 nickels and 20 dimes is $2.10.
3 nickels and 19 dimes is $2.05.
4 nickels and 18 dimes is $2.00.
5 nickels and 17 dimes is $1.95.
6 nickels and 16 dimes is $1.90.
7 nickels and 15 dimes is $1.85.
8 nickels and 14 dimes is $1.80.
9 nickels and 13 dimes is $1.75.
10 nickels and 12 dimes is $1.70.
11 nickels and 11 dimes is $1.65.
12 nickels and 10 dimes is $1.60.
13 nickels and 9 dimes is $1.55.
14 nickels and 8 dimes is $1.50.
15 nickels and 7 dimes is $1.45.
16 nickels and 6 dimes is $1.40.
17 nickels and 5 dimes is $1.35.
18 nickels and 4 dimes is $1.30.
19 nickels and 3 dimes is $1.25.
20 nickels and 2 dimes is $1.20.
21 nickels and 1 dime is $1.15.

Edwin