Understanding the Expression: $ (19a + 5b + c) - (7a + 3b + c) $

In mathematical modeling, simplifying algebraic expressions is essential for clarity and efficiency, especially when analyzing systems, solving equations, or optimizing algorithms. One commonly encountered expression is:

$$
(19a + 5b + c) - (7a + 3b + c)
$$

Understanding the Context

This article breaks down the simplification process, explores its meaning, and highlights how such algebraic manipulation supports broader applications in fields like engineering, economics, and computer science.

Step 1: Expanding the Expression

Begin by removing the parentheses, remembering that subtracting a sum is the same as subtracting each term inside:

$ (19a + 5b + c) - (7a + 3b + c)

Key Insights

$$
(19a + 5b + c) - 7a - 3b - c
$$

Now combine like terms:

  • For $ a $: $ 19a - 7a = 12a $
  • For $ b $: $ 5b - 3b = 2b $
  • For $ c $: $ c - c = 0 $

Thus, the simplified expression is:

$$
12a + 2b
$$

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Final Thoughts

Why Simplify This Expression?

At first glance, expanding a simple difference like this may seem trivial, but it reveals foundational skills:

  • Error reduction: Incorrect sign handling is a common mistake in algebra. Properly distributing the negative sign prevents sign errors.
  • Reduction of complexity: Combining like terms reduces clutter and uncovers the true relationship between variables.
  • Preparation for further analysis: Once simplified, $ 12a + 2b $ becomes easier to manipulate, plot, or input into models.

Practical Applications

The simplified form $ 12a + 2b $ frequently appears in:

  • Economics: Modeling cost functions where $ a $ and $ b $ represent units of different resources.
  • Machine learning: Feature weighting in linear regression, where coefficients ($ a, b, c $) weight input variables.
  • Engineering optimization: Evaluating system response based on variable magnitudes ($ a, b $) while constants ($ c $) offset baseline behavior.

Example: Applying the Simplified Form