Equation: 0.2x + 0.8(5 - x) = 0.5 * 5
Solving the Linear Equation: 0.2x + 0.8(5 - x) = 0.5 × 5 – A Step-by-Step Guide
Understanding linear equations is a fundamental skill in algebra, essential for students and professionals alike. One commonly encountered equation is:
0.2x + 0.8(5 - x) = 0.5 × 5
In this comprehensive article, we’ll break down this equation step-by-step, solve it clearly, and explore its real-world applications. Whether you're preparing for exams, teaching algebra, or solving practical problems, this guide will help you master the equation.
Understanding the Equation
The equation 0.2x + 0.8(5 - x) = 0.5 × 5 combines linear expressions involving a variable x and arithmetic operations. Let's rewrite and simplify each side to clarify the problem.
First, simplify the right-hand side: 0.5 × 5 = 2.5 So, the equation becomes: 0.2x + 0.8(5 - x) = 2.5
Step 1: Expand the Parentheses
Distribute 0.8 across (5 - x): 0.8 × 5 = 4 0.8 × (-x) = -0.8x Thus: 0.2x + (4 - 0.8x) = 2.5
Step 2: Combine Like Terms
Combine the x terms: 0.2x - 0.8x = -0.6x So: -0.6x + 4 = 2.5
Step 3: Isolate the Variable Term
Subtract 4 from both sides: -0.6x = 2.5 - 4 -0.6x = -1.5
Step 4: Solve for x
Divide both sides by -0.6: x = -1.5 / (-0.6) = 2.5
Final Answer
✅ x = 2.5
This solution confirms that when x = 2.5, substituting back into the original equation satisfies the equality.
Why This Equation Matters
Linear equations like this appear in real-life scenarios such as:
- Finance: Calculating break-even points where costs and revenue intersect.
- Physics: Determining points where velocities balance under different conditions.
- Engineering: Optimizing resource allocation under linear constraints.
Understanding how to set up and solve such equations gives learners strong analytical tools for both academic success and practical problem-solving.
Pro Tips for Solving Similar Equations
- Expand parentheses early to simplify expressions.
- Group like terms (terms with x and constants) on each side.
- Isolate the variable term carefully using inverse operations.
- Check your solution by plugging it back into the original equation.
- Practice with varied coefficients to build fluency.
Summary
Solving 0.2x + 0.8(5 - x) = 0.5 × 5 involves applying distributive property, combining like terms, and isolating x through basic algebra. With verification, we find:
x = 2.5
This represents a clear, step-driven understanding of linear equation solving—an essential foundation in mathematics.
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Want to master algebra? Keep practicing equations like this and expand your skills to tackle complex problems with confidence!
