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The image shows a set of four systems of equations, labeled (1) through (4). The task involves solving each system and determining the most efficient algebraic method for solving them.

### Let's analyze each system:

1. **System 1:**
   \[
   \begin{aligned}
   5x - 3y &= 19 \\
   2x - 6y &= 22
   \end{aligned}
   \]
   This is a system of two linear equations. A suitable method here would be either the *substitution* or *elimination* method. Because the second equation can easily be manipulated (with a factor) to eliminate \( y \), the *elimination* method seems efficient.

2. **System 2:**
   \[
   \begin{aligned}
   0.2x + 0.7y &= 1.5 \\
   0.3x + 0.2y &= 1
   \end{aligned}
   \]
   Another system of two linear equations, but this time with decimals. The best approach would be the *substitution* method after simplifying or multiplying the equations to avoid decimals.

3. **System 3:**
   \[
   \begin{aligned}
   y &= -6x \\
   2y &= 8x - 10
   \end{aligned}
   \]
   Here, the first equation directly gives \( y \) in terms of \( x \), making the *substitution* method the most straightforward way to solve.

4. **System 4:**
   \[
   \begin{aligned}
   \frac{x}{3} + \frac{y}{4} &= -1 \\
   2x + y &= -8
   \end{aligned}
   \]
   This system involves fractions, and the *elimination* method might be more efficient, especially by multiplying to eliminate the denominators first.

### a) For each system, the recommended methods:
1. *Elimination* method
2. *Substitution* method
3. *Substitution* method
4. *Elimination* method (after clearing fractions)

### b) Determining the solutions:
Would you like me to solve each system in detail?

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Here are some questions to explore further:
1. How does the elimination method work in solving linear systems?
2. What is the substitution method, and when is it most effective?
3. Why are decimals simplified in equations before solving?
4. How do you clear fractions in a system of equations?
5. How would you apply these methods to a 3-variable system?

**Tip:** Always simplify your system of equations before choosing a method. This makes calculations more efficient!

Solve the four systems of equations and determine the most efficient algebraic method for each system.

Explore how to solve four different systems of linear equations using the substitution and elimination methods. Learn the best strategies for simplifying equations and finding solutions in algebra, perfect for students in Grades 9-12.

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