Answer:
x = 23 $1 bills
y = 12 $5 bills
z = 3 $10 bills
w = 1 $20 bill
Step-by-step explanation:
First of all we have to setup linear equations with the help of given information.
x = number of $1 bills
y = number of $5 bills
z = number of $10 bills
w = number of $20 bills
The server has a total of $133 in denominations of $1, $5, $10, and $20 bills.
x + 5y + 10z + 20w = 133 eq. 1
The total number of paper bills is 39.
x + y + z + w = 39 eq. 2
The number of $5 bills is 4 times the number of $10 bills
y = 4z eq. 3
The number of $1 bills is 1 less than twice the number of $5 bills
x = 2y - 1 eq. 4
Now we have got 4 equations and four unknowns x, y, z, w
x + 5y + 10z + 20w = 133 eq. 1
x + y + z + w = 39 eq. 2
0 + y -4z + 0 = 0 eq. 3
x - 2y + 0 + 0 = -1 eq. 4
We have four equations and four unknowns x, y, z, w
We can solve this system of linear equations by various linear algebra methods like Crammer's rule, Gaussian Elimination etc. But if you notice eq. 3 and eq. 4 we have some zero entries in these equations and we might reduce the number of equations which will be easier and faster method to solve this system.
From eq. 3
0 + y -4z + 0 = 0
y = 4z
z = y/4
From eq. 4
x - 2y + 0 + 0 = -1
x = 2y - 1
Now put the values of x and z into eq. 1
x + 5y + 10z + 20w = 133
2y - 1 + 5y + 10(y/4) + 20w = 133
7y + (5/2)y + 20w = 133 + 1
9.5y + 20w = 134 eq. 5
Now put the values of x and z into eq. 2
x + y + z + w = 39
2y - 1 + y + y/4 + w = 39
3y + y/4 + w = 39 + 1
3.25y + w = 40 eq. 6
Now we have 2 equations ( eq. 5 and eq. 6) and 2 unknowns (y and w)
Solving these two equations simultaneously
9.5y + 20w = 134 eq. 5
3.25y + w = 40 eq. 6
Multiply eq. 6 by 20 and subtract it from eq. 5
9.5y + 20w = 134
65y + 20w = 800
-55.5y + 0 = -666
y = -666/-55.5 = 12
Put the value of y into eq. 5 to get the value of w
9.5(12) + 20w = 134
114 + 20w = 134
20w = 134 - 114
w = 20/20 = 1
Now we have got the value of y and we can substitute this value into eq. 3 and eq. 4 to get the values of x and z
From eq. 3
z = y/4 = 12/4 = 3
From eq. 4
x = 2y - 1 = 2(12) - 1 = 23
So we have solved the whole system of equations and got the following results
x = 23 $1 bills
y = 12 $5 bills
z = 3 $10 bills
w = 1 $20 bill
Verification:
Lets verify our results to check if we have got the correct numbers!
eq. 1 => x + 5y + 10z + 20w = 133
23 + 5(12) + 10(3) + 20(1) = 133
23 + 60 + 30 + 20 = 133
133 = 133 (hence satisfied)
eq. 2 => x + y + z + w = 39
23 + 12 + 3 + 1 = 39
39 = 39 (hence satisfied)
eq. 3 => 0 + y -4z + 0 = 0
0 + 12 -4(3) = 0
12 - 12 = 0
0 = 0 (hence satisfied)
eq. 4 => x - 2y + 0 + 0 = -1
23 - 2(12) + 0 + 0 = -1
23 - 24 = -1
-1 = -1 (hence satisfied)
