Title: Simplify and Solve: Solving the Equation (19a + 5b + c) − (7a + 3b + c) = 14

Introduction
Mastering algebraic equations is essential for students, educators, and anyone working with patterns and variables. One common type of problem involves simplifying expressions before solving. In this SEO-optimized guide, we explore the equation:

Understanding the Context

(19a + 5b + c) − (7a + 3b + c) = 25 − 11, and provide a step-by-step breakdown to solve it efficiently. With clear explanations, this article aims to help readers understand how to simplify complex expressions and arrive at accurate solutions—perfect for students preparing for algebra exams or anyone seeking clearer math comprehension.

Step-by-Step Breakdown of the Equation

Equation:
(19a + 5b + c) − (7a + 3b + c) = 25 − 11

\[ (19a + 5b + c) - (7a + 3b + c) = 25 - 11 \]

Key Insights

Step 1: Simplify the right-hand side
First, simplify the constant expression on the right:
25 − 11 = 14
So now the equation becomes:
(19a + 5b + c) − (7a + 3b + c) = 14

Step 2: Remove parentheses using the distributive property
When subtracting a parenthetical expression, distribute the negative sign:
= 19a + 5b + c − 7a − 3b − c

Step 3: Combine like terms
Group similar variables and constants:
(19a − 7a) + (5b − 3b) + (c − c)
= 12a + 2b + 0
= 12a + 2b

So now the equation simplifies to:
12a + 2b = 14

Step 4: Interpret the result
The equation has been reduced from involving three variables (a, b, c) to just two (a and b), meaning c cancels out completely—consistent with the original expression’s structure. This confirms the cancellation of c is valid and simplifies the solution pathway.

Continue Reading

$$ 4 = -a(0 - 3)^2 + 10 \implies 4 = 9a + 10 \implies 9a = -6 \implies a = - rac{2}{3}. $$
However, since the coefficient of $ x^2 $ is given as $ -a $, and parabolic arches open downward, $ a $ must be positive. Correcting the sign:
$$ 4 = -a(9) + 10 \implies -9a = 6 \implies a = - rac{2}{3} \quad ext{(invalid)}. $$

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Final Thoughts

Step 5: Express the final simplified equation
We now have:
12a + 2b = 14
This can be further simplified by dividing through by 2 for a cleaner form:
6a + b = 7

Why This Equation Matters for Students and Learners

This problem exemplifies a powerful algebraic principle: simplifying expressions before solving. By reducing terms and eliminating redundant variables like c, learners:

  • Improve their ability to approach complex equations confidently
  • Develop skills in organizing terms systematically
  • Enhance algebraic fluency, critical in higher math and STEM fields

Understanding how terms cancel and how coefficients simplify empowers students to tackle similar equations involving multiple variables and operations.

Final Solution & Key Takeaways

  • Original Equation: (19a + 5b + c) − (7a + 3b + c) = 25 − 11
  • Simplified Result: 12a + 2b = 14 → 6a + b = 7
  • Variable c disappears due to cancellation—showing that not all terms require individual solving
  • Purpose: Practice simplification and clarity in algebraic manipulation

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