$ (8a + 4b + 2c + d) - (a + b + c + d) = -1 - 3 $

Understanding the Simplified Equation: $ (8a + 4b + 2c + d) - (a + b + c + d) = -3 $

In algebra, simplifying expressions helps clarify hidden relationships and solve equations more effectively. One such expression commonly encountered is:

$$
(8a + 4b + 2c + d) - (a + b + c + d) = -3
$$

Understanding the Context

At first glance, the operation involves subtracting two polynomial expressions, but through step-by-step simplification, we uncover its true value and meaning.

Step-by-Step Simplification

Start with the original equation:

Key Insights

$$
(8a + 4b + 2c + d) - (a + b + c + d)
$$

Remove the parentheses by distributing the negative sign:

$$
8a + 4b + 2c + d - a - b - c - d
$$

Now combine like terms:

For $a$: $8a - a = 7a$
For $b$: $4b - b = 3b$
For $c$: $2c - c = c$
For $d$: $d - d = 0$

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Final Thoughts

So the simplified expression is:

$$
7a + 3b + c
$$

Thus, the equation becomes:

$$
7a + 3b + c = -3
$$

What Does This Mean?

The simplified equation shows a linear relationship among variables $a$, $b$, and $c$. While $d$ cancels out and does not affect the result, the final form reveals a constraint: the weighted sum $7a + 3b + c = -3$ must hold true.

This type of simplification is valuable in various applications, including:

Systems of equations — reducing complexity to isolate variables.
Optimization problems — identifying constraints in linear programming.
Algebraic reasoning — revealing underlying structure through elimination of redundant terms.