The problem involves two people, Emma and Madeline, and their savings over time. Let's break it down:

1. Emma's savings:
   Emma started with $65 and saves $10 each week. The total savings of Emma after $t$ weeks can be modeled by a linear equation: $E(t) = 65 + 10t$ where $E(t)$ represents the amount Emma has saved after $t$ weeks.

2. Madeline's savings:
   Madeline started with $25 and saves $15 each week. Her savings after $t$ weeks can be modeled by: $M(t) = 25 + 15t$ where $M(t)$ represents the amount Madeline has saved after $t$ weeks.

3. Comparing the savings:
   The next step is to determine when Emma has more savings than Madeline, which means solving for when $E(t) > M(t)$.

To solve this inequality: $65 + 10t > 25 + 15t$ Simplify it: $65 - 25 > 15t - 10t$ $40 > 5t$ Divide by 5: $8 > t$ or equivalently, $t < 8$

So, Emma has more savings than Madeline when $t < 8$.

Answer Summary:

• Emma's equation: $E(t) = 65 + 10t$
• Madeline's equation: $M(t) = 25 + 15t$
• Interval when Emma has more savings: $t < 8$

Would you like any further clarification or additional questions based on this? Here are some possible next steps:

1. What happens after 8 weeks? How do the savings compare?
2. How would the equations change if the savings rates were different?
3. What if Emma started with $70 instead of $65?
4. How could you graph these two equations to visualize the comparison?
5. How would the result change if Madeline saved only $5 per week?

Tip: When solving inequalities, always pay attention to the direction of the inequality when you divide or multiply by negative numbers.