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Let's go through the questions visible in the image and solve them accordingly.

### Question 9
Identify whether the system of equations has infinitely many solutions or no solution.

1. \( 4x + 8y = -8 \)
2. \( x - 2y = 1 \)

To determine if there are infinitely many solutions or no solution, let's examine if the equations are dependent (same line) or inconsistent (parallel lines).

1. Rewrite the first equation by dividing each term by 4:
   \[ x + 2y = -2 \]
2. Compare it with the second equation, \( x - 2y = 1 \).

These equations have opposite slopes, so they represent parallel lines, meaning **no solution** (they never intersect).

---

### Question 10
Identify whether the system of equations has infinitely many solutions or no solution.

1. \( 2x - 3y = 6 \)
2. \( 2x - 3y = -2 \)

Since both equations have the same left side (identical coefficients for \(x\) and \(y\)), but different constants on the right side, they are parallel and distinct lines. Thus, there is **no solution** (the lines do not intersect).

---

### Question 11
Identify whether the system of equations has infinitely many solutions or no solution.

1. \( 4x - 2y = 6 \)
2. \( 2x - y = 5 \)

The second equation is equivalent to the first equation divided by 2. However, when you try to solve these, they do not match (they have different constant terms on the right). So, these represent **parallel lines** with **no solution**.

---

### Question 12
Identify whether the system of equations has infinitely many solutions or no solution.

1. \( x - 5y = -5 \)
2. \( -x + 5y = -1 \)

Add the two equations:
\[
(x - 5y) + (-x + 5y) = -5 + (-1)
\]
\[
0 = -6
\]

Since the statement \( 0 = -6 \) is a contradiction, there is **no solution** to this system.

---

### Question 13
A hot air balloon problem:
- Mohamed's balloon is at an altitude of 40 m and rises 10 m/min.
- Dana’s balloon is at an altitude of 165 m and descends 15 m/min.

Let \( t \) be the time in minutes after they start.

**Equations:**
- Mohamed’s altitude: \( 40 + 10t \)
- Dana’s altitude: \( 165 - 15t \)

To find when they will be at the same altitude, set the equations equal to each other:
\[
40 + 10t = 165 - 15t
\]
Solving for \( t \):
\[
25t = 125 \Rightarrow t = 5 \, \text{minutes}
\]

**Altitude at that time:**
Substitute \( t = 5 \) into either equation:
\[
40 + 10(5) = 90 \, \text{meters}
\]

So, they will be at the same altitude of **90 meters after 5 minutes**.

---

### Question 14
Richard and Teo have a combined age of 31. Richard is 4 years older than twice Teo’s age.

Let \( x \) represent Teo's age and \( y \) represent Richard's age.

**Equations:**
1. \( x + y = 31 \)
2. \( y = 2x + 4 \)

Substitute \( y \) from the second equation into the first:
\[
x + (2x + 4) = 31
\]
\[
3x = 27 \Rightarrow x = 9
\]

Now, substitute \( x = 9 \) into the equation for \( y \):
\[
y = 2(9) + 4 = 22
\]

So, **Teo is 9 years old and Richard is 22 years old**.

---

Would you like further details or explanations on any of these solutions?

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### Follow-up Questions:
1. How do you identify whether a system of equations has infinitely many or no solutions?
2. How can you convert word problems into equations effectively?
3. What are some strategies for solving systems with substitution versus elimination?
4. Why do parallel lines in a system of linear equations indicate no solution?
5. How can understanding slopes help in analyzing solutions of linear systems?

### Tip:
When dealing with word problems, assign variables carefully and write down what each variable represents to avoid confusion.

Identify whether each system of equations has infinitely many solutions or no solution. Solve additional problems regarding altitude and age, as described in the image.

Explore how to identify solutions in systems of equations and solve real-life problems involving altitudes and ages. This guide covers the use of substitution, elimination, and identifying parallel lines, suitable for Grades 8-10.

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