2(2w + 3 + w) = 54 \quad \Rightarrow \quad 2(3w + 3) = 54
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.
Step 1: Simplify the Expression Inside the Parentheses
The equation begins with:
2(2w + 3 + w) = 54
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.
Step 2: Solve for w
Now, solve the simplified equation:
2(3w + 3) = 54
Divide both sides by 2To 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
$$w = \frac{24}{3} = 8$$
Final Answer
$$\boxed{w = 8}$$
Why This Equation Matters
Solving equations like 2(2w + 3 + w) = 54 helps build critical algebraic reasoning skills: distributing multiplying factors, combining like terms, and isolating variables. This type of problem appears frequently in math classes, standardized tests, and career fields that rely on analytical thinking.
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Conclusion
By simplifying the expression 2(2w + 3 + w) to 2(3w + 3), then isolating the variable through basic algebraic operations, you arrive easily at the solution w = 8. Mastering these steps strengthens your algebra foundation—essential for success in higher-level math and STEM fields.
If you’re learning algebra and often struggle with parentheses and distribution, remember: combine terms first, then simplify step by step. Practice makes perfect!
