w_2 + 2w_3 = -1 \quad ext(1) \ - Decision Point

Understanding the Context

What Does w₂ + 2w₃ = −1 ⁽¹⁾ Represent?

The equation w₂ + 2w₃ = −1 ⁽¹⁾ is a linear equation in two variables, typically w₂ and w₃. Here,

• w₂ and w₃ are variables representing unknown quantities.
  
• The coefficients (1 and 2) indicate how each variable contributes to the sum.
  
• The constant −1 is the value the left-hand side must equal.
  
• The superscript (1) often denotes a specific solution set or context, such as a system of equations.

This equation belongs to the realm of vector equations in algebra and serves as a building block for more complex multivariate systems.

Key Insights

Step-by-Step Interpretation

Let’s break down the equation:

1. Variables & Coefficients:
   

   • w₂ appears with coefficient 1, meaning its contribution is direct and unmodified.
     
   • w₃ appears with coefficient 2, indicating it has twice the weight in the sum.

2. Structure:
   The equation asserts that adding w₂ to twice w₃ yields −1.

3. System Context (if applicable):
   When paired with other equations (e.g., w₂ + 2w₃ = −1), this becomes part of a system of linear equations, useful in modeling real-world relationships—such as physics, economics, or engineering problems.

Final Thoughts

Solving the Equation: Tools and Techniques

To solve w₂ + 2w₃ = −1 ⁽¹⁾, follow these methods depending on your context:

1. Solving for One Variable

Express w₂ in terms of w₃:
w₂ = −1 − 2w₃
This means for any real number w₃, w₂ is uniquely determined, highlighting the equation’s dependence.

2. Graphical Interpretation

• Rewrite the equation as w₂ = −2w₃ − 1.
  
• Plot this linear relationship in the w₂–w₃ plane: it forms a straight line with:
  
  • Slope = −2 (steep downward line)
    
  • y-intercept at (0, −1) when w₃ = 0.

3. Using Vectors and Matrices

In linear algebra, this equation can be represented as:
[1 2] • [w₂ w₃]ᵀ = −1
Or as a matrix system:
Aw = b
Where:

• A = [1 2]
  
• w = [w₂; w₃]
  
• b = [−1]
  Solving involves techniques like Gaussian elimination or finding the inverse (if applicable in larger systems).