Cathy has to deposit $410 worth of five- and tendollar
bills. she has 1 fewer than three times as many
tens as she does five-dollar bills. how many of each bill
does she have to deposit?

(# of five dollar bills=f ; # of ten dollar bills=t)
3t-1=f ⇒ this means that if there were 1 $5 bills, there would be 2 $10 bills.
So the ratio of $5 bills to $10 bills is 1 to 2.
We can plug that into another equation:
f+t=410 ⇒ I havent actually learned this in school yet so I just made a chart:         $10 ║ $5
║
2     ║ 1    = $25
4     ║ 2    = $50
6     ║ 3    = $75
8     ║ 4    = $100
10   ║ 5    = $125
and so on...
you probably found the pattern already, so you can conclude that [cathy deposited 16 five dollar bills and 33 ten dollar bills]

Also, you should probably check if there are any mistakes when going through this since I took almost an hour writing this.