$ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $

February 22, 2026

Simplifying the Equation: $ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $

Understanding algebraic expressions can sometimes feel complex, but breaking down equations step-by-step makes them far more manageable. Today, we’ll explore the equation $ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $ and simplify it intelligently.

Step 1: Rewrite the subtraction as distribution

The left-hand side involves subtraction of two binomials. Distribute the negative sign across the second group:

$$ (37a + 7b + c) - 19a - 5b - c $$

Now combine like terms:

For $ a $: $ 37a - 19a = 18a $
For $ b $: $ 7b - 5b = 2b $
For $ c $: $ c - c = 0 $

So the simplified left side becomes: $$ 18a + 2b $$

Step 2: Simplify the right-hand side

The right-hand side is a constant expression:

$$ -12 - 6 = -18 $$

So now the equation looks like: $$ 18a + 2b = -18 $$

Step 3: Simplify further

We can factor the left-hand side by pulling out the common factor 2:

$$ 2(9a + b) = -18 $$

Divide both sides by 2: $$ 9a + b = -9 $$

What does the simplified equation mean?

We’ve transformed: $$ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $$ into: $$ 9a + b = -9 $$

This shows that regardless of the values of $ c $ (which canceled out during subtraction), the result depends only on $ a $ and $ b $. The equation now emphasizes a linear relationship between $ a $ and $ b $.

Why does $ 37a + 7b + c - (19a + 5b + c) = -12 - 6 $ work?

Subtracting $ c $ from $ c $ eliminates it, simplifying the comparison. The remaining terms reflect net changes in $ a $ and $ b $, matching the right side — a constant (-18). This confirms that $ a $ and $ b $ must adjust precisely to satisfy the equation.

Practical application

If you’re solving real-world problems—like balancing equations in chemistry, economics, or algebra—these simplifications help isolate relationships and solve for unknowns efficiently.

Final Thoughts

Breaking $ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $ step-by-step reveals a clean, solvable relationship: $$ 9a + b = -9 $$

Remember: canceling identical terms (here $ c $) reduces complexity and highlights key variables. Mastering such simplifications strengthens foundational algebra skills essential for advanced math.

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Meta Description: Learn how to simplify and solve the equation $ (37a + 7b + c) - (19a + 5b + c) = -12 - 6 $ step-by-step, revealing the simplified form $ 9a + b = -9 $ and practical algebraic insights. Perfect for algebra students and teachers.