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

Determine the missing steps and justifications in her work.
[tex]
\begin{array}{l}
y - 2,000 = 2,000 + 320x - 2,000 \quad \text{division property of equality} \quad \frac{y - 2,000}{320} = x \\
\frac{10,000 - 2,000}{320} = 25 = x \quad \frac{8,000 - 2,000}{320} = 24 = x \quad \frac{y - 320}{2,000} = x
\end{array}
[/tex]
division property of equality
[tex]
y - 2,000 = 2,000 - 320x - 2,000
[/tex]
multiplication property of equality
\begin{tabular}{|c|l|}
\hline
\multicolumn{1}{|c|}{ Steps } & \multicolumn{1}{|c|}{ Justification } \\
\hline
$y = 2,000 + 320x[tex]$ & given \\
\hline
$[/tex]y - 2,000 = 320x[tex]$ & subtraction property of equality \\
\hline
$[/tex]\frac{y - 2,000}{320} = x$ & division property of equality \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Here is the detailed, step-by-step solution and justification for Stella's question:

1. Given Equation:
- Step: [tex]\( y = 2000 + 320x \)[/tex]
- Justification: This is the initial equation provided by Stella to represent her situation, where [tex]\( y \)[/tex] is the total amount in her emergency fund, [tex]\( 2000 \)[/tex] is her initial amount, and [tex]\( 320x \)[/tex] is the amount she saves every month for [tex]\( x \)[/tex] months.

2. Subtract 2000 from both sides:
- Step: [tex]\( y - 2000 = 2000 + 320x - 2000 \)[/tex]
- Justification: Subtraction property of equality. We subtract 2000 from both sides to isolate the term involving [tex]\( x \)[/tex].

3. Simplify:
- Step: [tex]\( y - 2000 = 320x \)[/tex]
- Justification: Simplify the equation by combining like terms on the right-hand side.

4. Divide by 320:
- Step: [tex]\( \frac{y - 2000}{320} = x \)[/tex]
- Justification: Division property of equality. Divide both sides by 320 to solve for [tex]\( x \)[/tex].

Now, applying the given values:

5. Substitute [tex]\( y = 10000 \)[/tex]:
- Step: [tex]\( \frac{10000 - 2000}{320} = x \)[/tex]
- Justification: Substitute the goal amount of [tex]$10,000 into the equation. 6. Calculate: - Step: \( \frac{8000}{320} = 25 \) - Justification: Perform the division to determine the number of months. 7. Result: - Step: \( x = 25 \) - Justification: Final result after calculation. It takes Stella 25 months to reach her goal of $[/tex]10,000 in her emergency fund.

Thus, all of the steps and justifications align with the conclusion that Stella will need to save for 25 months to reach her $10,000 goal.

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