Solve the system of equations to find the total amount of money Dawn has in each account.

Dawn has been using two bank accounts to save money for a car. The difference between account 1 and account 2 is [tex]$\$[/tex]100[tex]$. 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]\[tex]$2,000$[/tex].

[tex]\[
\begin{array}{l}
A - B = 100 \\
\frac{3}{8} A + \frac{7}{8} B = 2000
\end{array}
\][/tex]

Multiply each equation by a number that produces opposite coefficients for [tex]$A$[/tex] or [tex]$B$[/tex]:

[tex]\[
\begin{array}{l}
A - B = 100 \\
\frac{3}{8} A + \frac{7}{8} B = 2000
\end{array}
\][/tex]



Answer :

Sure, let’s solve this problem step-by-step.

Problem Statement:
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 \(\frac{3}{8}\) of account 1 and \(\frac{7}{8}\) of account 2, Dawn will have a down payment of $[/tex]2000. Find the total amount of money Dawn has in each account.

Equations to solve:
[tex]\[ A - B = 100 \][/tex]
[tex]\[ \frac{3}{8}A + \frac{7}{8}B = 2000 \][/tex]

Step-by-Step Solution:

1. Equation 1:
[tex]\[ A - B = 100 \][/tex]

2. Equation 2:
[tex]\[ \frac{3}{8}A + \frac{7}{8}B = 2000 \][/tex]

To eliminate the fractions in Equation 2, multiply the entire equation by 8:
[tex]\[ 8 \left( \frac{3}{8}A + \frac{7}{8}B \right) = 8 \cdot 2000 \][/tex]
Simplifying, we get:
[tex]\[ 3A + 7B = 16000 \][/tex]

Now, we have a new simpler system of equations:
[tex]\[ A - B = 100 \][/tex]
[tex]\[ 3A + 7B = 16000 \][/tex]

3. Solving the system using substitution or elimination:

First, solve Equation 1 for [tex]\(A\)[/tex]:
[tex]\[ A = B + 100 \][/tex]

Next, substitute [tex]\(A = B + 100\)[/tex] into the second equation:
[tex]\[ 3(B + 100) + 7B = 16000 \][/tex]
[tex]\[ 3B + 300 + 7B = 16000 \][/tex]
[tex]\[ 10B + 300 = 16000 \][/tex]

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

Divide by 10:
[tex]\[ B = 1570 \][/tex]

Now, substitute [tex]\(B = 1570\)[/tex] back into [tex]\(A = B + 100\)[/tex]:
[tex]\[ A = 1570 + 100 \][/tex]
[tex]\[ A = 1670 \][/tex]

Therefore, Dawn has \[tex]$1670 in account 1 and \$[/tex]1570 in account 2.