Understanding the Perimeter Equation: \( 2(w + 3w) = 48 \)

Calculating the perimeter of geometric shapes starts with understanding key expressions and solving equations that describe them. One common algebraic challenge involves perimeter formulas in terms of a variable. Consider the equation:

\[2(w + 3w) = 48\]

This equation represents a real-world scenario where you’re working with the perimeter of a shape whose side lengths involve a variable \( w \), multiplied by constants. Solving it yields the value of \( w \), which helps determine the actual size of the perimeter.

Simplifying the Expression Inside the Parentheses

The expression \( w + 3w \) represents the sum of two related side lengths—possibly adjacent sides of a rectangle or similar figure. Combining like terms:

\[w + 3w = 4w\]

Substitute this back into the original equation:

\[2(4w) = 48\]

Solving for \( w \)

Now simplify:

\[8w = 48\]

To isolate \( w \), divide both sides by 8:

\[w = \frac{48}{8} = 6\]

Thus, the value of \( w \) is 6.

Determining the Perimeter

Now that you know \( w = 6 \), plug it into the expression \( w + 3w = 4w \):

\[4w = 4 \ imes 6 = 24\]

The full perimeter is twice this sum, as per the formula:

\[\ ext{Perimeter} = 2(w + 3w) = 2 \ imes 24 = 48\]

This confirms the solution satisfies the original equation.

Real-Life Application and Key Takeaways

This equation models situations where a shape’s perimeter depends on a single variable measured in units (like meters or feet). Whether analyzing a rectangular boundary or planning fencing for a construction site, solving \( 2(w + 3w) = 48 \) helps find the value needed to construct or verify a space.

Key takeaways:

• Combine like terms: Simplify expressions inside parentheses before multiplying.- Isolate the variable: Divide both sides by the coefficient to solve for \( w \).- Verify the solution: Substitute back to ensure the perimeter equals the given value.

Final Answer

\[w = 6\]

\[\ ext{Perimeter} = 48 \ ext{ units}\]

Understanding and solving equations like \( 2(w + 3w) = 48 \) strengthens algebraic skills and applies directly to geometry, architecture, engineering, and everyday problem solving.

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