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To answer question 12, let’s set up the system of equations for the situation:

1. **Define variables:**
   - Let \( t \) be the time in minutes.
   - Let \( h_M(t) \) and \( h_D(t) \) be the altitudes of Mohamed's and Dana's balloons at time \( t \), respectively.

2. **Equation for Mohamed's balloon:**
   - Starting altitude: 40 m
   - Rising rate: 10 m/min
   - Equation: \( h_M(t) = 40 + 10t \)

3. **Equation for Dana's balloon:**
   - Starting altitude: 165 m
   - Descending rate: -15 m/min
   - Equation: \( h_D(t) = 165 - 15t \)

### Part (a)
To find when both balloons are at the same altitude, set \( h_M(t) = h_D(t) \):

\[
40 + 10t = 165 - 15t
\]

Solve for \( t \):

1. Combine terms: \( 25t = 125 \)
2. Divide by 25: \( t = 5 \)

So, both balloons will be at the same altitude after **5 minutes**.

### Part (b)
To find the altitude at this time, substitute \( t = 5 \) into either equation:

Using \( h_M(t) = 40 + 10t \):
\[
h_M(5) = 40 + 10(5) = 40 + 50 = 90 \text{ m}
\]

Thus, the altitude will be **90 meters**.

---

### Question 13 Solution
For the next problem:
- **Define variables for ages:**
   - Let \( x \) be Teo's age.
   - Richard's age would then be \( 2x + 4 \).

- Given that their combined age is 31:
   \[
   x + (2x + 4) = 31
   \]

   Simplify and solve:
   \[
   3x + 4 = 31
   \]
   \[
   3x = 27
   \]
   \[
   x = 9
   \]

   So, Teo is **9 years old**, and Richard is:
   \[
   2(9) + 4 = 18 + 4 = 22 \text{ years old}
   \]

---

Would you like a more detailed explanation, or have further questions?

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### Additional Questions for Practice:
1. What if Mohamed's balloon started at a different altitude? How would that change the equations?
2. How would the answer change if Dana's balloon descended at a rate of 10 m/min instead of 15 m/min?
3. If both balloons were rising, how would the system of equations differ?
4. How would you approach solving for age if Richard was 3 years older than twice Teo’s age instead of 4?
5. What are the steps to verify the answers in both problems?

### Tip:
Always start by defining variables and setting up clear equations for each condition in a word problem.

Write and solve a system of equations for the situation at a hot air balloon festival and determine Richard and Teo's ages.

Solve real-world systems of equations involving altitude and age problems. Learn how to find when two hot air balloons reach the same altitude and calculate Richard and Teo's ages using algebra. Suitable for Grades 8-10.

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