Question

Dawn has been using two bank accounts to save money for a car. The difference between account 1 and account 2 is $100. If she uses 3/8 of account 1 and 7/8 of account 2, Dawn will have a down payment of $2,000. Solve the system of equations to find the total amount of money Dawn has in each account.

A – B = 100

Three-eighthsA + Seven-eighthsB = 2,000

Dawn has $
in account 1 and $
in account 2.

Answer 1

Let's define the variables for the amounts in each account:

- Let \( A \) be the amount in account 1.
- Let \( B \) be the amount in account 2.

We have the following system of equations:

1. \( A - B = 100 \)
2. \( \frac{3}{8}A + \frac{7}{8}B = 2000 \)

Now we can solve this system step-by-step.

### Step 1: Solve the first equation for \( A \):

From the first equation,
\[
A = B + 100
\]

### Step 2: Substitute \( A \) in the second equation:

We can substitute \( A \) in the second equation:
\[
\frac{3}{8}(B + 100) + \frac{7}{8}B = 2000
\]

### Step 3: Distribute and combine like terms:

Distributing the \( \frac{3}{8} \):
\[
\frac{3}{8}B + \frac{300}{8} + \frac{7}{8}B = 2000
\]

Combine the terms involving \( B \):
\[
\left(\frac{3}{8}B + \frac{7}{8}B\right) + \frac{300}{8} = 2000
\]
\[
\frac{10}{8}B + \frac{300}{8} = 2000
\]
This simplifies to:
\[
\frac{5}{4}B + \frac{300}{8} = 2000
\]

### Step 4: Multiply the entire equation by 8 to eliminate fractions:

Multiply by 8:
\[
10B + 300 = 16000
\]

### Step 5: Solve for \( B \):

Subtract 300 from both sides:
\[
10B = 15700
\]

Divide by 10:
\[
B = 1570
\]

### Step 6: Substitute back to find \( A \):

Now we can find \( A \) using the equation \( A = B + 100 \):
\[
A = 1570 + 100 = 1670
\]

### Summary of Results

Dawn has:

- \( A = 1670 \) in account 1
- \( B = 1570 \) in account 2

Thus, the final amounts are:
- Dawn has **$1670** in account 1 and **$1570** in account 2.