3/4x + 10 = 7/8x.
Understanding and Solving the Equation: 3/4x + 10 = 7/8x
Mathematics often presents challenges, especially when solving for variables in linear equations. One common type of problem involves isolating the variable x in expressions with fractions. In this article, we’ll explore how to solve the equation 3/4x + 10 = 7/8x step by step, explain the key algebraic concepts, and provide practical tips for correcting such equations—whether you're doing homework, preparing for exams, or simply expanding your math skills.
Solving the Equation: Step-by-Step Guide
Step 1: Understand the equation
We start with: 3/4x + 10 = 7/8x
Our goal is to isolate x on one side. This requires manipulating both sides using inverse operations while preserving equality.
Step 2: Eliminate the fractions by finding a common denominator
Both coefficients of x—3/4 and 7/8—are fractions. To simplify, express them with a common denominator (8):
- 3/4 = 6/8
- 7/8 already has denominator 8
Rewriting the equation: (6/8)x + 10 = (7/8)x
Step 3: Subtract (6/8)x from both sides
This isolates the constant on the left and terms with x on the right: 10 = (7/8)x – (6/8)x 10 = (1/8)x
Step 4: Solve for x
Now solve: x = 10 × 8 x = 80
Final Answer
The solution to 3/4x + 10 = 7/8x is x = 80
Why This Equation Matters: Key Concepts
- Linear Equations: This is a first-degree equation—meaning x appears only to the first power—common in algebra and physics applications.
- Fractions & Common Denominators: Converting fractions to a common denominator simplifies arithmetic and avoids errors.
- Balancing Techniques: Any operation (addition, subtraction, multiplication) applied to one side must be applied to the other to maintain equality.
Verifying the Solution
Substitute x = 80 back into the original equation:
Left: 3/4 × 80 + 10 = 60 + 10 = 70
Right: 7/8 × 80 = 70
Both sides equal 70, confirming our solution is correct.
Tips for Mastering Such Equations
- Always rewrite fractions using common denominators—this avoids common mistakes in addition and subtraction.
- Organize terms with like coefficients—group x terms on one side and constants on the other.
- Check your work—substitution prevents weak conclusions and reinforces understanding.
- Practice varied problem types—mixes of fractions, decimals, and whole numbers build flexibility.
Real-World Applications
Linear equations like 3/4x + 10 = 7/8x appear in budgeting (e.g., calculating break-even points), physics (e.g., motion equations), and engineering where ratios and proportions are key.
Conclusion
Solving 3/4x + 10 = 7/8x teaches essential algebra skills: working with fractions, balancing equations, isolating variables, and verifying solutions. With consistent practice and clear steps, solving linear equations becomes intuitive and powerful. Start today—strengthening your math foundation is never a waste of time!
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By mastering this equation and its solution method, you’re building strong analytical skills useful far beyond the classroom. Keep practicing, stay curious, and let math empower your problem-solving journey!
