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:

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

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