Perimeter: \( 2(w + 3w) = 48 \) - Decision Point
Understanding the Perimeter Equation: \( 2(w + 3w) = 48 \)
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:
\[
2(w + 3w) = 48
\]
Understanding the Context
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:
\[
w + 3w = 4w
\]
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Key Insights
Substitute this back into the original equation:
\[
2(4w) = 48
\]
Solving for \( w \)
Now simplify:
\[
8w = 48
\]
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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.
