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

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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.