How would you work, and solve this problem?
Christa is a waitress and collects her tips at the table. At the end of the shift she has 68 bills, all ones and fives. If the total value of her tips is 172, how many of each bill does she have?
n = number of ones
1n = value of ones
68 - n = number of fives
5(68 - n) = value of fives
n + 5(68 - n) = 172
n + 340 - 5n = 172
-4n = -168
n = 42
n = 42 ones = 42.00
68 - n = 26 fives = 130.00
68 bills = 172
To solve this problem, we can set up a system of linear equations. Let's use variables to represent the number of one-dollar bills and five-dollar bills Christa has.
Let's call the number of one-dollar bills x and the number of five-dollar bills y.
We know that Christa has a total of 68 bills, so we can create our first equation: x + y = 68.
We also know that the total value of her tips is $172. Since she has x one-dollar bills and y five-dollar bills, we can create our second equation: 1x + 5y = 172.
Now, we can solve this system of equations to find the values of x and y.
One way to solve it is by substitution:
1. Solve the first equation for x:
x = 68 - y.
2. Substitute the value of x in the second equation:
1(68 - y) + 5y = 172.
3. Simplify and solve for y:
68 - y + 5y = 172,
68 + 4y = 172,
4y = 172 - 68,
4y = 104,
y = 26.
4. Substitute the value of y back into the first equation to find x:
x + 26 = 68,
x = 42.
So, Christa has 42 one-dollar bills and 26 five-dollar bills.
In summary, to solve this problem, we set up a system of equations using variables to represent the number of one-dollar bills and five-dollar bills. We then solve the system using substitution to find the values of x and y, which give us the number of each bill Christa has.