SOLUTION: You must state your answer in words. You must set up a system of two equations in two variables. You must state what the variables represent. You must sh

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Question 193326: You must state your answer in words. You must set up a system of two equations in two variables.
You must state what the variables represent.
You must show all the steps in solving the system of equations.
Stan has a collection of nickels and dimes that is worth $6.05. If the number of dimes was doubled and the number of nickels was decreased by 10, the value of the coins would be $9.85. How many dimes does he have? Thanks!
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Let d represent the number of dimes. Let n represent the number of nickels.

The value of each dime is 10 cents, so the value of the dimes in the collection is 10d. Likewise, the value of the nickels is 5n. Since we have established the value of the coins in cents, convert the total value of the collection from dollars and cents to just cents -- i.e. move the decimal point. $6.05 equals 605 cents.

Value equation: The value of the dimes added to the value of the nickels must equal the value of the collection.



Double the number of dimes is 2d, so the value of double the number of dimes must be 20d. Decreasing the number of nickels by 10 is n - 10, so the value of the new amount of nickels must be .

"What if" value equation: The value of the new amount of dimes plus the value of the new amount of nickels must equal value of the "what if" collection (again, expressed in cents):



Put this second equation into standard form:



Multiply this equation by -1:



Add this equation to the first value equation:





Notice that the n variable has been eliminated by choosing a multiplier for one of the equations such that the coefficients on n became additive inverses.



Substitute this value for d into either original value equation; let's use the first one:







So the collection contains 43 dimes worth $4.30, and 35 nickels worth $1.75. $4.30 plus $1.75 is $6.05. Double the number of dimes is 86, worth $8.60 and ten less nickels is 25, worth $1.25, $8.60 plus $1.25 is $9.85 so the answer checks.


John