Question 104903
The number of coins is 7. 
The total value of the coins is $0.85.
You have quarters, nickels, and dimes. 
You must use at least one of each coin. 
Let'assign a name to each coin. 

Q-quarters
N-nickels
D-dimes
The number of coins is 7.
1.{{{Q+N+D=7}}}
Now let's look at the value of each coin and the total. 
2.{{{0.25Q+0.05N+0.10D=0.85}}}
I have to use each coin once. 
{{{Q>=1}}}
{{{N>=1}}}
{{{D>=1}}}
We don't have enough information since we have two equations and three unknowns. 
Let's see if we can go through logically and determine an answer.
Use equation 1. 
1.{{{Q+N+D=7}}}
{{{N=7-Q-D}}}
Substitute into equation 2.
2.{{{0.25Q+0.05N+0.10D=0.85}}}
{{{25Q+5N+10D=85}}} Multiply by 10 to simplify the decimals.
{{{25Q+5(7-Q-D)+10D=85}}} Substitute.
{{{25Q+35-5Q-5D+10D=85}}}
{{{20Q+5D=50}}}
Let's solve for D in terms of Q. 
{{{20Q+5D=50}}}
{{{5D=50-20Q}}}
3.{{{D=10-4Q}}}
Let's use our inequality equations now. 
{{{N>=1}}}
{{{7-Q-D>=1}}}
{{{-Q-D>=-6}}}
{{{Q+D>=6}}}
From 3, 
{{{Q+D<=6}}}
{{{Q+(10-4Q)<=6}}}
{{{10-3Q<=6}}}
{{{-3Q<=-4}}}
{{{Q>=4/3}}}
You also know from above
{{{D>=1}}}
{{{10-4Q>=1}}}
{{{-4Q>=-9}}}
{{{Q<=9/4}}}
Now you know that 
{{{4/3<=Q<=9/4}}}
Q must take on integer values since it's a coin, then you conclude, 
{{{Q=2}}}
From 3,
3.{{{D=10-4Q}}}
{{{D=10-4(2)}}}
{{{D=2}}}
And finally,
{{{N=7-Q-D}}}
{{{N=7-2-2}}}
{{{N=3}}}
2 Quarters, 2 Dimes, and 3 Nickels.
Check your answer. 
2*25+2*10+3*5=85
50+20+15=85
85=85
Good answer.