1. Consider the problem: Assume you have two credit cards charging the equivalent of $1.75 \%$ and $2.25 \%$ simple interest per month. If your total 'interest-bearing' balance for both cards is $\$ 5,402.89$ and you are being charged $\$ 104.18$ in interest for one month, what is your 'interest-bearing' balance on each card? a. Explain why this application can be classified as a mixture application.

Step 1 ：Consider the problem: Assume you have two credit cards charging the equivalent of $1.75 \%$ and $2.25 \%$ simple interest per month. If your total 'interest-bearing' balance for both cards is $\$ 5,402.89$ and you are being charged $\$ 104.18$ in interest for one month, what is your 'interest-bearing' balance on each card?

Step 2 ：This problem can be classified as a mixture application because it involves two different 'ingredients' or 'components' (the two credit cards) that are mixed together to form a total (the total 'interest-bearing' balance). The 'mixture' here is the total balance, and the 'components' are the balances on each card. The problem involves finding the amount of each component given the total amount of the mixture and the 'rate' or 'proportion' of each component (the interest rates).

Step 3 ：To solve this problem, we can set up a system of linear equations. The first equation will represent the total balance, and the second equation will represent the total interest. We can then solve this system to find the balance on each card.

Step 4 ：Let's denote the balance on the first card as x and the balance on the second card as y. Then we have: 1) \(x + y = 5402.89\) (total balance) 2) \(0.0175x + 0.0225y = 104.18\) (total interest)

Step 5 ：We can solve this system of equations to find the values of x and y.

Step 6 ：Final Answer: The 'interest-bearing' balance on the first card is \(\boxed{x}\) dollars and on the second card is \(\boxed{y}\) dollars.