SOLUTION: A bag of coins contains 57 nickels and dimes. The value of coins is $3.95 a) define the variable b) state the system of equations that models this situation c) determine t

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Question 1011836: A bag of coins contains 57 nickels and dimes. The value of coins is $3.95
a) define the variable
b) state the system of equations that models this situation
c) determine the number of nickels and dimes
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
Let the number of nickels be x
Let the number of dimes be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
nickels       x      $0.05     $0.05x
dimes         y      $0.10     $0.10y
-------------------------------------------
TOTALS       57      -----     $3.95

 The first equation comes from the "Number of coins"
 column.

  %28matrix%283%2C1%2CNumber%2Cof%2Cnickels%29%29%22%22%2B%22%22%28matrix%283%2C1%2CNumber%2Cof%2Cdimes%29%29%22%22=%22%22%28matrix%284%2C1%2Ctotal%2Cnumber%2Cof%2Ccoins%29%29

                  x + y = 57

 The second equation comes from the last column.
  %28matrix%284%2C1%2CValue%2Cof%2CALL%2Cnickels%29%29%22%22%2B%22%22%28matrix%284%2C1%2CValue%2Cof%2CALL%2Cdimes%29%29%22%22=%22%22%28matrix%285%2C1%2CTotal%2Cvalue%2Cof%2CALL%2Ccoins%29%29

           0.05x + 0.10y = 3.95

Get rid of decimals by multiplying every term by 100:

               5x + 10y = 395

 So we have the system of equations:
           system%28x+%2B+y+=+57%2C5x+%2B+10y+=+395%29.

We solve by substitution.  Solve the first equation for y:

           x + y = 57
               y = 57 - x

Substitute (57 - x) for y in 5x + 10y = 395

    5x + 10(57 - x) = 395
     5x + 570 - 10x = 395
          -5x + 570 = 395
                -5x = -175
                  x = 35 = the number of nickels.

Substitute in y = 57 - x
              y = 57 - (35
              y = 22 dimes.


Checking:  35 nickels is $1.75 and 22 dimes is $2.20
            That's 57 coins.
            And indeed $1.75 + $2.20 = $3.95
Edwin