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Solving and Understanding the Equation: x + ½x = 5 – A Complete Guide

When faced with the equation x + ½x = 5, it’s easy to feel overwhelmed, especially for learners new to algebra. However, this straightforward expression is a gateway into solving linear equations and understanding key algebraic concepts. In this article, we’ll explore step-by-step how to solve for x, dive into why this equation works, and highlight practical applications of this simple yet powerful formula.

Understanding the Context

What is the Equation: x + ½x = 5?

At first glance, the equation x + ½x = 5 appears deceptively simple. It combines a variable x with half of itself, set equal to 5. Algebraically, this is a linear equation — one of the foundational building blocks in mathematics — and solving it helps develop critical problem-solving skills.

Step-by-Step Guide to Solving x + ½x = 5

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Key Insights

Step 1: Combine Like Terms

Start by simplifying the left-hand side. Since x can be written as 1x, the equation becomes:
x + ½x = 1x + 0.5x = 1.5x

So now, the equation reads:
1.5x = 5

Step 2: Isolate the Variable

To solve for x, divide both sides by 1.5:
x = 5 ÷ 1.5

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Final Thoughts

Step 3: Perform the Division

Recall that dividing by a decimal can be tricky. To simplify:
5 ÷ 1.5 = (5 × 10) ÷ (1.5 × 10) = 50 ÷ 15

Now divide:
x = 50 / 15 = 10 / 3 ≈ 3.333...

Final Answer

The solution to the equation x + ½x = 5 is:
x = 10⁄3

Understanding the Mathematics Behind the Equation

To grasp the significance of x + ½x = 5, let’s explore what this represents.

x + ½x = (1 + 0.5)x = 1.5x, which reflects a proportional relationship.
This equation models real-world situations where a quantity is half of itself plus the full quantity — for example, when combining two parts of a whole.
The result x = 10⁄3 shows that x is not an integer, emphasizing that algebraic solutions often involve fractional or irrational numbers.