SOLUTION: I know how to solve a linear equation with three variables by using two of the three equations to eliminate one variable and then use a different pair of equations to eliminate the

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Question 142764: I know how to solve a linear equation with three variables by using two of the three equations to eliminate one variable and then use a different pair of equations to eliminate the same variable and then you can solve those equations. I am doing some word problems and writing the first two equations with three unknown variables is easy but I cannot seem to write the third.
Mike has 19 coins (nickels, dimes and quarters) in his piggy bank with a total value of $2.35. There are 2 more dimes than nickels. How many of each type does Mike have?
N+D+Q=19
5N+10D+25Q=235
?

With the first two I can multiply the first by –5 to eliminate N. I need another equation. Help?
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Fred, Tiger and Nancy play a round of golf together. Their combined score is 223. Fred’s score was 3 more than Nancy’s and Tiger’s score was 7 more than Fred’s. What was each person’s score.
You would think something simple like
F=N+3
T=F+7
223=T+F+N
But I need three equations with three unknown variables.
Found 2 solutions by stanbon, checkley77:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Mike has 19 coins (nickels, dimes and quarters) in his piggy bank with a total value of $2.35. There are 2 more dimes than nickels. How many of each type does Mike have?
N+D+Q=19
5N+10D+25Q=235
?
Comment:
"There are 2 more dimes than nickels."
D = N+2
or
N -D -0 = -2
This is your 3rd equation
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Fred, Tiger and Nancy play a round of golf together. Their combined score is 223. Fred’s score was 3 more than Nancy’s and Tiger’s score was 7 more than Fred’s. What was each person’s score.
You would think something simple like
F=N+3
T=F+7
223=T+F+N
But I need three equations with three unknown variables.
------------
T + F + N = 223
0 + F - N = 3
T - F + 0 = 7
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These are your three equations.
Cheers,
Stan H.

Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
N+D+Q=19
5N+10D+25Q=235
D=N+2 IS THE THIRD EQUATION.
N+(N+2)+Q=19
2N+2+Q=19
2N+Q=19-2
2N+Q=17 MULTIPLY THIS EQUATION BY -25 & ADD IT TO THE [15N+25Q=215] EQUATION
5N+10(N+2)+25Q=235
5N+10N+20+25Q=235
15N+25Q=235-20
15N+25Q=215
-50N-25Q=-425
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-35N=-210
N=-210/-35
N=6 ANSWER FOR THE NUMBER OF NICKELS.
D=6+2
D=8 ANSWER FOR THE NUMBER OF DIMES.
2*6+Q=17
12+Q=17
Q=17-12
Q=5 NUMBER OF QUARTERS.
PROOF:
6+8+5=19
19=19
AND:
5*6+10*8+25*5=235
30+80+125=235
235=235
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YOU HAVE THE THREE REQUIRED EQUATION HOWEVER YOU DO NOT NEED ALL THREE VARIABLES IN EACH EQUATION. iN FACT IT IS EASIER TO SOLVE IF THERE ARE ONLY TWO VARIABLES.
F=N+3
T=F+7 OR T=(N+3)+7 OR T=N+10
223=T+F+N OR 223=(N+10)+(N+3)+N
223=N+10+N+3+N
223=3N+13
3N=223-13
3N=210
N=210/3
N=70 FOR NANCY'S SCORE.
F=70+3=73 FOR FRED'S SCORE.
T=73+7=80 FOR TIGER'S SCORE.
PROOF:
223=70+73+80
223=223