Question

Write a system of linear equations from the following situation and select the correct solution. Ava has 35 paper money bills in her wallet, all five-dollar bills and twenty-dollar bills. The total value of the money is $415. How many of each type of bill does Ava have? (1 point) Responses 21 five-dollar bills and 14 twenty-dollar bills 21 five-dollar bills and 14 twenty-dollar bills 13 five-dollar bills and 22 twenty-dollar bills 13 five-dollar bills and 22 twenty-dollar bills 19 five-dollar bills and 16 twenty-dollar bills 19 five-dollar bills and 16 twenty-dollar bills 16 five-dollar bills and 19 twenty-dollar bills

Answer 1

To formulate a system of linear equations based on the situation described for Ava's bills, we can define two variables:

Let:
- \( x \) = the number of five-dollar bills
- \( y \) = the number of twenty-dollar bills

From the information given, we can create the following equations:

1. The total number of bills:
\[
x + y = 35
\]

2. The total value of the bills:
\[
5x + 20y = 415
\]

Now we can solve this system of equations.

First, we can simplify the second equation by dividing every term by 5:
\[
x + 4y = 83
\]

Now we have the system of equations:
1. \( x + y = 35 \)
2. \( x + 4y = 83 \)

Next, we can use the first equation to substitute \( x \) in the second equation:
From \( x + y = 35 \), we can express \( x \) as:
\[
x = 35 - y
\]

Now, substituting this expression for \( x \) into the second equation:
\[
(35 - y) + 4y = 83
\]
\[
35 - y + 4y = 83
\]
\[
35 + 3y = 83
\]
\[
3y = 83 - 35
\]
\[
3y = 48
\]
\[
y = 16
\]

Now that we have \( y \), we can find \( x \):
\[
x = 35 - y = 35 - 16 = 19
\]

So, Ava has:
- \( x = 19 \) five-dollar bills
- \( y = 16 \) twenty-dollar bills

Thus, the solution is:
**19 five-dollar bills and 16 twenty-dollar bills.**

Comparing this with the available responses, the correct answer is:
**19 five-dollar bills and 16 twenty-dollar bills.**