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9:No Solution
Infinite Solutions
One Solution
10:No Solution
Infinite Solutions
One Solution

The image contains two linear equations:

1. \( 4x + 8y = -8 \)
2. \( 2x - 3y = 6 \)

Let's analyze each pair to determine if there is a unique solution, infinitely many solutions, or no solution.

### Step 1: Standardize and Compare the Equations

**Equation 1:**
\[
4x + 8y = -8
\]
Dividing by 4 to simplify:
\[
x + 2y = -2
\]

**Equation 2:**
\[
2x - 3y = 6
\]

Now, we have the system:
\[
x + 2y = -2
\]
\[
2x - 3y = 6
\]

### Step 2: Determine Consistency and Solve the System

1. **Substitute or Eliminate**: Let's use substitution or elimination to see if there’s a solution.
2. **Solving by Elimination**: Multiply the first equation by 2 to align the \( x \) terms:
   \[
   2(x + 2y) = 2(-2) \Rightarrow 2x + 4y = -4
   \]

   Now we have:
   \[
   2x + 4y = -4
   \]
   \[
   2x - 3y = 6
   \]

3. **Subtract the Equations**:
   \[
   (2x + 4y) - (2x - 3y) = -4 - 6
   \]
   \[
   7y = -10
   \]
   \[
   y = -\frac{10}{7}
   \]

4. **Find \( x \)**: Substitute \( y = -\frac{10}{7} \) back into \( x + 2y = -2 \):
   \[
   x + 2\left(-\frac{10}{7}\right) = -2
   \]
   \[
   x - \frac{20}{7} = -2
   \]
   \[
   x = -2 + \frac{20}{7}
   \]
   \[
   x = \frac{6}{7}
   \]

### Conclusion:
The system has **one unique solution**:
\[
(x, y) = \left(\frac{6}{7}, -\frac{10}{7}\right)
\]

**Answers:**
- **Problem 9**: One Solution
- **Problem 10**: One Solution

Would you like more details or have any questions?

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### Related Questions:

1. How do we determine when a system of equations has no solution?
2. What conditions lead to a system having infinitely many solutions?
3. How can we use matrix methods to solve systems of equations?
4. How does substitution work in systems of linear equations?
5. Can a system of linear equations have more than one unique solution?

**Tip:** Simplifying equations by dividing all terms can make solving systems much easier.

9: No Solution, Infinite Solutions, One Solution
10: No Solution, Infinite Solutions, One Solution

Analyze systems of linear equations to determine if there is a unique solution, infinitely many solutions, or no solution. This includes step-by-step methods using elimination and substitution to solve the system and understand consistency.

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