Solution: Rewrite in slope-intercept form: $ 3y = 2x + 12 $ → $ y = rac23x + 4 $. The $ y $-intercept occurs at $ x = 0 $, so $ y = 4 $. - Deep Underground Poetry

Understanding the Context

Start with the given linear equation:
$$ 3y = 2x + 12 $$

To rewrite it in slope-intercept form, divide every term by 3 to isolate $ y $:
$$
y = rac{2x + 12}{3}
$$
$$
y = rac{2}{3}x + 4
$$

Identifying the $ y $-Intercept

Key Insights

The slope-intercept form reveals key features immediately:

• The slope $ m = rac{2}{3} $, which indicates that for every 3 units increase in $ x $, $ y $ increases by 2.
  
• The $ y $-intercept $ b = 4 $, meaning the line crosses the $ y $-axis at $ (0, 4) $.

This point is critical because it gives a clear starting point for graphing or analyzing the equation’s behavior.

Why This Matters: Practical Applications

Rewriting equations in slope-intercept form is foundational in algebra, making it easier to:

• Graph linear relationships clearly
  
• Compare slopes and intercepts between different equations
  
• Solve real-world problems involving linear trends, such as predicting costs, measuring growth, or analyzing data linear relationships

Final Thoughts

Summary: Key Takeaway

To transform any line equation into slope-intercept form:

1. Isolate $ y $ by dividing all terms by the coefficient of $ y $.
   
2. Simplify the right-hand side.
   
3. Identify $ m $ and $ b $ clearly for interpretation.

In the example $ 3y = 2x + 12 $, solving step-by-step delivers the clean $ y = rac{2}{3}x + 4 $ form—providing both slope and $ y $-intercept at a glance. Mastery of this technique strengthens your algebraic toolkit and supports advanced math topics ahead.

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