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.

Understanding the Context

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.

3/4x + 10 = 7/8x.

Key Insights

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

Continue Reading

This means $ x + 3 $ is a common multiple of $ 7, 8, $ and $ 9 $.
First, compute the least common multiple (LCM) of $ 7, 8, $ and $ 9 $:
\text{lcm}(7, 8, 9) = \text{lcm}(7, \text{lcm}(8, 9)) = \text{lcm}(7, 72) = 504

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Final Thoughts

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