Understanding the Equation: $ (64a + 16b + 4c + d) - (27a + 9b + 3c + d) = -7 - 5 $

When tackling algebraic expressions, simplifying and analyzing equations plays a crucial role in both academic learning and real-world applications. In this article, we break down the expression:

$$
(64a + 16b + 4c + d) - (27a + 9b + 3c + d) = -7 - 5
$$

Understanding the Context

and solve it step-by-step to reveal its hidden value and meaning.

Step 1: Simplify the Left-Hand Side (LHS)

We begin by simplifying the expression using the distributive property:

$ (64a + 16b + 4c + d) - (27a + 9b + 3c + d) = -7 - 5 $

Key Insights

$$
(64a + 16b + 4c + d) - (27a + 9b + 3c + d)
$$

Distribute the minus sign across the second parentheses:

$$
64a + 16b + 4c + d - 27a - 9b - 3c - d
$$

Now combine like terms:

  • $a$-terms: $64a - 27a = 37a$
  • $b$-terms: $16b - 9b = 7b$
  • $c$-terms: $4c - 3c = 1c = c$
  • $d$-terms: $d - d = 0$

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Final Thoughts

So the simplified LHS becomes:

$$
37a + 7b + c
$$

Step 2: Simplify the Right-Hand Side (RHS)

The right-hand side is:

$$
-7 - 5 = -12
$$

Step 3: Set Up the Equation

Now the equation reads:

$$
37a + 7b + c = -12
$$