Use a system of equations to solve problem

Question

Emily emptied her bank ana totaled the nickels, dimes, and quarters that she had saved. The total amount was 14.50
The number of dimes was 2 more than 3 times the number of nickels. How many of each coin were there

Can you show the steps i just cant do this can you show every step so i can understand this

Answer 1

Let N be number of nickels, D dimes, and q quarters.

VAlue equation:
.25q+ .10d + .05n=14.50

d=3N+2

.25q+.10(3N+2)+.05n=14.50
.25q+.30n +.20+.05n=14.50
.25q+.35n=14.30

Now q, n have to be integers...you cant have a fraction of a coin.
So lets get the nickels out of the way first by getting the quarter total value to some multiple of a dollar, or 25c, or 50 cents...
How many .35 will it take to make the total a multiple of .25? Turns out, three...If n is 3, then .35n=1.05, so
.25q=14.30-1.05=13.25 so q is you can do that.

Answer 2

let the number of nickels, dimes and quarters be N, D, and Q
but D= 3N+2

5N + 10(3N+2) + 25Q = 1450
25Q + 35N = 1430
5Q + 7N = 286

we want integer solutions for N and Q
by some trial and error I got
Q=25
and N = 23, so D = 71

check: 25*25 + 71*10 + 23*5 = 1450

My question to you "Is this answer unique" ?