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:

Solved: 21. (a+2b+3c)-(4a+6b-5c) is equivalent to: A. -4a-8b-2c B. -4a ...

Image Gallery

Solved: 21. (a+2b+3c)-(4a+6b-5c) is equivalent to: A. -4a-8b-2c B. -4a ...
Solved Is the set S = {(7a -5b + c,a -b + c,c-a, b + c,a, b | Chegg.com
Solved: 4a+a+5+7a+2 5b-b-7+b c+8c-2c+5b 3a^2+6a+a^2 [Math]
Solved 17) 6a + b + c = 8 -6a + b + c = -20 -3a + 4b - 3c = | Chegg.com
3a+b+c=7 a+3b-c=13 -2a+b+OC=-1 [algebra]

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:

$$
12a + 2b
$$

Continue Reading

\times 7 = 210
Therefore, the greatest common divisor of 420 and 630 is \(\boxed{210}\).### Question 1
A cylindrical water tank has a radius of 3 meters and a height of 10 meters. Calculate the volume of the tank in cubic meters. Use the formula for the volume of a cylinder: \( V = \pi r^2 h \).

🔗 Related Articles You Might Like:

📰 Media Player Videolan 📰 Happymod Apk 📰 Extensiones Chrome 📰 Eptinezumab 6992721 📰 Fios Apps For Tv 270063 📰 Girls Frontline 1 The Ultimate Uncut Campaign That Will Blow Your Socks Off 9340923 📰 Nude Male Muscular 7007446 📰 Credit Cards 0 Apr 6603421 📰 Master Billiards In 30 Dayswatch This Beginner Break Records 193628 📰 Define Grittier 9902129 📰 Gad Scoring 4284976 📰 Figma Ai 8958551 📰 Huang He The Lifeline Ancient Civilizations Fought For But Never Trusted 8182340 📰 The Truth About Khalid Jordans Latest Masterpieceit Changed Everything 3101408 📰 Download The Java Development Kit For Windows Fastget It Free Today 5960045 📰 Can Mosquitoes Bite Through Clothes 154475 📰 Apple Tv New Movies 188153 📰 Princess Cut Engagement Ring Secret The Most Sought After Design That Weds Dream And Brilliance 9443865

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