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.

Understanding the Context

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:

Solved 17) 6a + b + c = 8 -6a + b + c = -20 -3a + 4b - 3c = | Chegg.com

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Solved 17) 6a + b + c = 8 -6a + b + c = -20 -3a + 4b - 3c = | Chegg.com
Solved c) 3A+4B+5C+6D2B+A+4D+3C2C−A8D+6C−A+4B=7=5=3=1 d) | Chegg.com
Solved Is the set S = {(7a -5b + c,a -b + c,c-a, b + c,a, b | Chegg.com
Solved: a) 4a+7b+5a-b b) 6c+4d-c-7d [Math]
Solved: a) 4a+7b+5a-b b) 6c+4d-c-7d [Math]

Key Insights

  • 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:

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Final Thoughts

$$
-12 - 6 = -18
$$

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:
$$
9a + b = -9
$$