A bank teller is counting the total amount of money in a cash drawer at the end of the shift. There is a total of $3416 in denominations of $1, $5, $10, and $20 bills. The total number of paper bills is 307. The number of $20 bills is twice the number of $1 bills, and the number of $5 bills is 10 more than the number of $1 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination. Step 1 To find the equations for the given model, first define the variables as given below. x = number of $1 bills y = number of $5 bills z = number of $10 bills w = number of $20 bills As per the given conditions, the system of linear equations will be as follows. + 5y + 10z + 20w = 3416 + y + W = 307 - 2x w = -2r - x + y 10 Write the associated augmented matrix for the system. 1 5 10 20 : 3416 1 1 1 1 : 307 -2 0 1: -1 1. 10 ...

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A bank teller is counting the total amount of money in a cash drawer at the end of the shift. There is a total of
$3416 in denominations of $1, $5, $10, and $20 bills. The total number of paper bills is 307. The number of
$20 bills is twice the number of $1 bills, and the number of $5 bills is 10 more than the number of $1 bills.
Write a system of linear equations to represent the situation. Then use matrices to find the number of each
denomination.
Step 1
To find the equations for the given model, first define the variables as given below.
x = number of $1 bills
y = number of $5 bills
z = number of $10 bills
w = number of $20 bills
As per the given conditions, the system of linear equations will be as follows.
+ 5y + 10z + 20w = 3416
+ y +
z +
W =
307
- 2x
W =
-2r
- x
+ y
10
Write the associated augmented matrix for the system.
15 10 20 3416
1 :
1 1
307
-2
1:
-1
1.
10
Transcribed Image Text:A bank teller is counting the total amount of money in a cash drawer at the end of the shift. There is a total of $3416 in denominations of $1, $5, $10, and $20 bills. The total number of paper bills is 307. The number of $20 bills is twice the number of $1 bills, and the number of $5 bills is 10 more than the number of $1 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination. Step 1 To find the equations for the given model, first define the variables as given below. x = number of $1 bills y = number of $5 bills z = number of $10 bills w = number of $20 bills As per the given conditions, the system of linear equations will be as follows. + 5y + 10z + 20w = 3416 + y + z + W = 307 - 2x W = -2r - x + y 10 Write the associated augmented matrix for the system. 15 10 20 3416 1 : 1 1 307 -2 1: -1 1. 10
Step 2
To write the matrix in reduced row-echelon form, apply elementary row operations until we obtain zeros
above and below each of the leading 1's.
First interchange the rows of the augmented matrix as given below.
- 20 0
- 1 1
1 1
15 10 20 3416
1:
10
1.
307
Now perform the operation (-R1)
R1.
on
(-글씨)-
R1-
1 0
-1 1
10
1:
307
15 10
20 : 3416
Next, perform the operations (R2 + R1
on R2 and (R3 - R1) on R3.
0 0
글 :
글 :
1
10
(R2 + R1)→
(R3 - R1)→
1
1
307
1
5 10
20 : 3416
Transcribed Image Text:Step 2 To write the matrix in reduced row-echelon form, apply elementary row operations until we obtain zeros above and below each of the leading 1's. First interchange the rows of the augmented matrix as given below. - 20 0 - 1 1 1 1 15 10 20 3416 1: 10 1. 307 Now perform the operation (-R1) R1. on (-글씨)- R1- 1 0 -1 1 10 1: 307 15 10 20 : 3416 Next, perform the operations (R2 + R1 on R2 and (R3 - R1) on R3. 0 0 글 : 글 : 1 10 (R2 + R1)→ (R3 - R1)→ 1 1 307 1 5 10 20 : 3416
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