Bill has 30 coins, some nickels some dimes. They equal $2.10. How many nickels are there? What is the equasion used to solve this problem?

Let d=# dimes.
let n=# nickels.

d + n = 30
d($.10) + n($.05) = $2.10

Two equations and two unknowns. Solve them simultaneously.

To solve this problem, we can use a system of equations, which involves finding the intersection point of two lines. In this case, we have two unknowns: the number of dimes (d) and the number of nickels (n).

The first equation represents the total number of coins: d + n = 30. This equation states that the number of dimes plus the number of nickels equals 30, which is the total number of coins Bill has.

The second equation represents the total value of the coins: d($0.10) + n($0.05) = $2.10. This equation states that the value of the dimes (d multiplied by $0.10) plus the value of the nickels (n multiplied by $0.05) equals $2.10, which represents the total value of all the coins.

To solve this system of equations simultaneously, you can use one of several methods such as substitution, elimination, or graphical methods. Let's use the substitution method for this example:

We start by solving the first equation for one variable in terms of the other. Rearranging the equation, we get d = 30 - n.

Now, substitute this expression for d in the second equation: (30 - n)($0.10) + n($0.05) = $2.10.

Expanding the equation, we get 3 - 0.10n + 0.05n = $2.10.

Combine like terms: 0.05n - 0.10n = $2.10 - $3.

Simplifying, we get -0.05n = - $0.90.

Dividing both sides of the equation by -0.05, we find that n = $0.90 / (-0.05) = 18.

So, there are 18 nickels.

To find the number of dimes, substitute this value for n in the first equation: d + 18 = 30.

Solving for d, we get d = 30 - 18 = 12.

Therefore, there are 12 dimes.

To summarize, the equation used to solve this problem is:

d + n = 30 (equation 1)
0.10d + 0.05n = 2.10 (equation 2)

By using substitution, we find that there are 18 nickels and 12 dimes.