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Stella wants to build up her emergency fund to [tex]$\$[/tex]10,000[tex]$. She has $[/tex]\[tex]$2,000$[/tex] now and plans on saving [tex]$\$[/tex]320[tex]$ every month. To model the situation, she creates this equation, where $[/tex]x[tex]$ represents the number of months and $[/tex]y[tex]$ represents the total amount in her emergency fund:
\[ y = 2,000 + 320x \]

Determine the missing steps and justifications in her work.
\[
\begin{array}{l}
y - 2,000 = 2,000 + 320x - 2,000 \\
\frac{y - 2,000}{320} = x \\
\frac{10,000 - 2,000}{320} = 25 = x \\
\frac{8,000 - 200}{320} = 24 = x \\
\frac{y - 320}{2,000} = x \\
y - 2,000 = 2,000 - 320x - 2,000 \\
\end{array}
\]
\[ \text{division property of equality} \]
\[ \text{multiplication property of equality} \]

\begin{tabular}{|l|l|}
\hline
Steps & Justification \\
\hline
$[/tex]y = 2,000 + 320x[tex]$ & Given equation \\
\hline
$[/tex]y - 2,000 = 320x[tex]$ & Subtraction property of equality \\
\hline
$[/tex]\frac{y - 2,000}{320} = x[tex]$ & Division property of equality \\
\hline
$[/tex]\frac{10,000 - 2,000}{320} = 25 = x$ & Substitution and simplification \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's analyze the steps and justifications in Stella's work.

Steps and Justifications Table:
```plaintext
| Steps | Justification |
|------------------------------------------|------------------------------------------|
| y = 2,000 + 320 x | Given equation |
| y - 2,000 = 2,000 + 320 x - 2,000 | Subtraction Property of Equality |
| y - 2,000 = 320 x | Simplification |
| (y - 2,000) / 320 = x | Division Property of Equality |
| (10,000 - 2,000) / 320 = x | Substitution and Simplification |
| 25 = x | Simplification |
```

Explanation of each step:

1. Given Equation:
[tex]\( y = 2,000 + 320 x \)[/tex]

2. Subtraction Property of Equality:
To isolate [tex]\( x \)[/tex], subtract [tex]$2,000 from both sides of the equation: \[ y - 2,000 = 2,000 + 320 x - 2,000 \] 3. Simplification: Simplify the equation to: \[ y - 2,000 = 320 x \] 4. Division Property of Equality: To solve for \( x \), divide both sides by $[/tex]320:
[tex]\[ \frac{y - 2,000}{320} = x \][/tex]

5. Substitution and Simplification:
Replace [tex]\( y \)[/tex] with the target emergency fund amount 10,000:
[tex]\[ \frac{10,000 - 2,000}{320} = x \][/tex]

6. Simplification:
Perform the simplification:
[tex]\[ \frac{8,000}{320} = 25 \][/tex]
Therefore:
[tex]\[ x = 25 \][/tex]

Stella needs to save for 25 months to reach her goal of $10,000.

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