Okay, let's see. The problem is that Hillary has eleven coins, all dimes and nickels, and the total value is 75 cents. I need to find out how many nickels she has. Hmm, let me think about how to approach this.

First, I remember that dimes are worth 10 cents and nickels are 5 cents each. So, if I let the number of dimes be D and the number of nickels be N, then the total number of coins is D + N = 11. And the total value is 10D + 5N = 75 cents. That makes sense.

So, I can set up a system of equations here. Let me write that down:

1. D + N = 11
2. 10D + 5N = 75

Now, I need to solve this system of equations. Maybe I can use substitution or elimination. Let me try substitution. From the first equation, I can express D in terms of N. So, D = 11 - N. Then, substitute this into the second equation.

Substituting D into the second equation: 10*(11 - N) + 5N = 75. Let's compute that step by step.

First, expand the terms: 10*11 is 110, and 10*(-N) is -10N. So, 110 - 10N + 5N = 75. Combine like terms: -10N + 5N is -5N. So, 110 -5N = 75.

Now, subtract 110 from both sides: -5N = 75 - 110. That's -5N = -35. Then, divide both sides by -5: N = (-35)/(-5) = 7. So, N is 7. That means there are 7 nickels.

Wait, let me check that. If there are 7 nickels, then the number of dimes is 11 -7 = 4. So, 4 dimes would be 4*10 = 40 cents, and 7 nickels would be 7*5 = 35 cents. Adding those together: 40 + 35 = 75 cents. That matches the total. So, that seems correct.

Alternatively, I could have used elimination. Let's try that method to verify. The second equation is 10D +5N =75. If I multiply the first equation by 5, I get 5D +5N =55. Then subtract the first equation multiplied by 5 from the second equation:

(10D +5N) - (5D +5N) =75 -55. That gives 5D =20. So, D=4. Then N=11-4=7. Same result. So, that confirms it.

Another way to think about it is using the concept of averages or total value. Since all coins are either dimes or nickels, maybe calculate how much each coin type contributes. But the algebraic method seems straightforward here.

Let me also check if there are other possible solutions. Suppose there were more nickels. For example, if N=8, then D=3. Then total value would be 3*10 +8*5=30+40=70, which is less than 75. If N=6, D=5: 5*10 +6*5=50+30=80, which is more than 75. So, 7 nickels is the only number that works. So, that's the answer.
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