Solving the Equation: 2(2w + 3 + w) = 54 → Simplifying to 2(3w + 3) = 54

Understanding and solving linear equations is a fundamental skill in algebra that forms the basis of more advanced mathematics. One common type of equation you often encounter is when parentheses are used with expressions involving variables. In this article, we’ll walk through step-by-step how to solve the equation:

2(2w + 3 + w) = 54,
and show how it simplifies neatly to:
2(3w + 3) = 54,
leading to an easy solution for w.

Understanding the Context

Step 1: Simplify the Expression Inside the Parentheses

The equation begins with:

2(2w + 3 + w) = 54

$\Rightarrow \quad \frac{a}{b}=\frac{3+1+2 \sqrt{3}}{3+1-2 \sqrt{3}} \Righ..$

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$\Rightarrow \quad \frac{a}{b}=\frac{3+1+2 \sqrt{3}}{3+1-2 \sqrt{3}} \Righ..$

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Key Insights

Before solving, combine like terms inside the parentheses:

2w + w = 3w
- The constant +3 remains.

So,
2w + 3 + w = 3w + 3

Now the equation becomes:
2(3w + 3) = 54

This matches the simplified form 2(3w + 3) = 54, as shown.

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Final Thoughts

Step 2: Solve for w

Now, solve the simplified equation:

2(3w + 3) = 54

Divide both sides by 2
To isolate the parenthetical expression, divide both sides by 2:

$$
\frac{2(3w + 3)}{2} = \frac{54}{2}
\Rightarrow 3w + 3 = 27
$$

Subtract 3 from both sides

$$
3w + 3 - 3 = 27 - 3
\Rightarrow 3w = 24
$$

Divide both sides by 3

$$
w = \frac{24}{3} = 8
$$