Answer :
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.
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.