Define the variable and equation:
lauren's piggy bank contains 141 coins made up of nickels and dimes. the total value of the coins is $0.85. how many nickels and dimes are there?

-- Each nickel is worth $0.05, so if there are 'N' nickels, their value is  0.05N .

-- Each dime is worth 0.10, so if there are 'D' dimes, their value is 0.10D .

-- We know that  (N + D) = (number of nickels + number of dimes) = 141

You later edited your question, and said the total value in the pig
is actually $10.85,  so ...
 
-- We also know that (0.05N + 0.10D) =
                                 (value of the nickels+value of the dimes) = 10.85

Now we have two equations ... exactly what we need when there are
two unknown quantities to be found.

                                            N +       D   =  141
                                     0.05N + 0.10D  =   10.85

Multiply each side of the
second equation by  10 :        0.5N + D = 108.5

Now your two equations
look like this:                        N + D = 141
                                       0.5N + D = 108.5

Subtract the second one
from the first one:              0.5N + 0  =  32.5

Multiply this one by  2 :          N  =  65          There are 65 nickels in the pig.

How about the dimes ?  That's easy now.  You know there are
141 coins in there all together, so the number of dimes is just   
                                                                                           (141 - 65) = 76 dimes.



Check:

         Total value = 0.05N  +  0.10D

         What's the total value if  N=65  and  D = 76  ?

         0.05(65) + 0.10(76) = 3.25 + 7.60 = 10.85        It checks.              yay !