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.