Let's solve the given system of equations step by step:
The two equations are:
[tex]\[
\begin{aligned}
1. & \quad A - B = 100 \\
2. & \quad \frac{3}{8}A + \frac{7}{8}B = 2000
\end{aligned}
\][/tex]
### Step 1: Solve the first equation for [tex]\(A\)[/tex]
We can rewrite the first equation in terms of [tex]\(A\)[/tex]:
[tex]\[ A = B + 100 \][/tex]
### Step 2: Substitute this expression into the second equation
By substituting [tex]\(A = B + 100\)[/tex] into the second equation, we get:
[tex]\[ \frac{3}{8}(B + 100) + \frac{7}{8}B = 2000 \][/tex]
### Step 3: Simplify and solve for [tex]\(B\)[/tex]
First, distribute [tex]\(\frac{3}{8}\)[/tex] within the parentheses:
[tex]\[ \frac{3}{8}B + \frac{3}{8} \cdot 100 + \frac{7}{8}B = 2000 \][/tex]
This simplifies to:
[tex]\[ \frac{3}{8}B + \frac{3}{8} \cdot 100 + \frac{7}{8}B = 2000 \][/tex]
Combine the terms involving [tex]\(B\)[/tex]:
[tex]\[ \left(\frac{3}{8} + \frac{7}{8}\right)B + \frac{3}{8} \cdot 100 = 2000 \][/tex]
Since [tex]\(\frac{3}{8} + \frac{7}{8} = 1\)[/tex]:
[tex]\[ B + \frac{3}{8} \cdot 100 = 2000 \][/tex]
Calculate [tex]\(\frac{3}{8} \cdot 100\)[/tex]:
[tex]\[ B + 37.5 = 2000 \][/tex]
Isolate [tex]\(B\)[/tex]:
[tex]\[ B = 2000 - 37.5 \][/tex]
[tex]\[ B = 1962.5 \][/tex]
### Step 4: Solve for [tex]\(A\)[/tex] using the expression from Step 1
Substitute [tex]\(B = 1962.5\)[/tex] back into the equation [tex]\(A = B + 100\)[/tex]:
[tex]\[ A = 1962.5 + 100 \][/tex]
[tex]\[ A = 2062.5 \][/tex]
Therefore, Dawn has [tex]\(\$2062.5\)[/tex] in account 1 and [tex]\(\$1962.5\)[/tex] in account 2.
