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:

If a: b :: c: d, show that 2a +3b 2a- 3b: 2c+ 3d 2c 3d. a:b: c: d a/b = c/d

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If a: b :: c: d, show that 2a +3b 2a- 3b: 2c+ 3d 2c 3d. a:b: c: d a/b = c/d
Solved a) 2a + 2b-c+d=4 4a + 3b - C + 2d = 6 8a + 5b - 3c + | Chegg.com
Solved b.-3a +4b - 6c = -2 4a +8b+2c - d 4 2a - 6b +2c + 4d | Chegg.com
Solved a) 2a+2b−c+d=44a+3b−c+2d=68a+5b−3c+4d=123a+3b−2c+2d=6 | Chegg.com
Solved c) 3A+4B+5C+6D2B+A+4D+3C2C−A8D+6C−A+4B=7=5=3=1 d) | Chegg.com

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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#### Range: 35.36 m**Question:
Compute the product of the roots of the equation \( v\sqrt{v} - 5v + 6\sqrt{v} = 0 \), given that all of the roots are real and positive.
First, let \( \sqrt{v} = t \), then \( v = t^2 \). Substitute into the equation:

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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.