Question

A food server examines the amount of money earned in tips after working an 8-hour shift. the server has a total of $133 in denominations of $1, $5, $10, and $20 bills. the total number of paper bills is 39. the number of $5 bills is 4 times the number of $10 bills, and the number of $1 bills is 1 less than twice the number of $5 bills. write a system of linear equations to represent the situation. (assume x = number of $1 bills, y = number of $5 bills, z = number of $10 bills, and w = number of $20 bills.)

Answer 1

Here is a system of linear equations that represents the situation.

x +5y +10z +20w = 133 . . . total amount earned

x +y +z +w = 39 . . . . . . . . . total number of bills

y = 4z . . . . . . . . . . . . . . . . . . the number of 5s is 4 times the number of 10s

x = 2y -1 . . . . . . . . . . . . . . . . the number of 1s is 1 less than twice the number of 5s

_____

We can substitute for x and z in the first two equations:

... (2y-1) +5y +10(y/4) +20w = 133

... (2y-1) +y +(y/4) +w = 39

These simplify to

... 9.5y +20w = 134

... 3.25y +w = 40

Solving by your favorite method, you get

... y = 12

... w = 1

So the other values can be found to be

... x = 2·12 -1 = 23

... z = 12/4 = 3

The solution to the system is (x, y, z, w) = (23, 12, 3, 1).