How to Solve Perimeter Problems Like Perimeter = 2(x + 3x) = 8x = 64 → x = 8: A Step-by-Step Guide

Understanding perimeter problems is essential for mastering geometry, especially in algebra. Whether you’re solving math homework or tackling real-world measurement challenges, knowing how to simplify expressions and isolate variables is key. In this article, we’ll explore how solving the equation Perimeter = 2(x + 3x) = 8x = 64 → x = 8 works step-by-step, helping you confidently handle similar perimeter problems.

Understanding the Context

Understanding the Perimeter Equation

In geometry, the perimeter of a shape is the total distance around its edges. For polygons such as rectangles, perimeter can be calculated using formulas based on side lengths.

In our example:
Given the perimeter expression:
Perimeter = 2(x + 3x) = 8x
and Perimeter = 64, we set up the equation:
8x = 64

This equation tells us that eight times a variable x equals 64. Our goal is to find the value of x, which represents a fundamental length in this shape.

Perimeter = 2(x + 3x) = 8x = 64 → x = 8.

Key Insights

Step 1: Simplify the Expression Inside Parentheses

Start with:
2(x + 3x)

First, simplify inside the parentheses:
x + 3x = 4x

Now the equation becomes:
2(4x) = 64

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Question: A micropaleontologist applies Fourier-like decomposition to periodic microfossil abundance data, solving $\mathbf{v} \times \mathbf{u} = \mathbf{w}$, where $\mathbf{u} = \langle 1, 2, 3 \rangle$, $\mathbf{w} = \langle 4, -2, 1 \rangle$. If $\mathbf{v} = \langle a, b, c \rangle$, determine the rank of the homogeneous system arising from the cross product.
A: 1
B: 0

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Final Thoughts

Which simplifies to:
8x = 64

Step 2: Solve for x

To isolate x, divide both sides of the equation by 8:
8x ÷ 8 = 64 ÷ 8
x = 8

This shows that the value of x is 8 — a crucial piece of information for finding actual side lengths.

Step 3: Verify the Solution

Plug x = 8 back into the original expression to confirm:
Perimeter = 2(x + 3x) = 2(8 + 3×8) = 2(8 + 24) = 2×32 = 64

The math checks out — the perimeter is indeed 64 when x = 8.