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

[tex]\[
\begin{array}{l}
A - B = 100 \\
\frac{3}{8}A + \frac{7}{8}B = 2,000
\end{array}
\][/tex]

Multiply each equation by a number that produces opposite coefficients for [tex]\(A\)[/tex] or [tex]\(B\)[/tex].

Sagot :

GinnyAnswer
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.
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