Solving the Equation:

7 ⁻⁴⁄₃ + 3·(²⁵⁄²) + c = –4 ▶ Interpreting and Solving Linear Equation

When tackling algebra, simplifying and solving linear equations is a fundamental skill. One such problem involves simplifying irrational and fractional components to isolate an unknown variable, c. This article breaks down the equation step-by-step and provides a clear explanation for solving 7 – ⁴⁄³ + 3 · (²⁵⁄²) + c = –4, confirming its validity.

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Understanding the Equation

We begin with: 7 – ⁴⁄³ + 3 · (²⁵⁄²) + c = –4

This equation contains:

• A rational mixed number: 7 and ⁴⁄³
• An irrational component: ²⁵⁄² (which simplifies to a decimal or fraction)
• A constant coefficient: 3 multiplied by the irrational term
• An unknown constant c, which we must isolate

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Step-by-Step Simplification

Step 1: Convert Mixed Number to Improper Fraction

7 can be written as an improper fraction: 7 = 7/1 7 = (7 × 3)/3 = 21/3 To combine with ⁴⁄³, convert both to a common denominator (denominator = 3): 21/3 – ⁴⁄3 = (21 – 4)/3 = 17/3

Now the equation becomes: 17⁄3 + 3·(²⁵⁄²) + c = –4

Step 2: Multiply the Fractional Multiplier

Calculate 3 × (²⁵⁄²): 3 × ²⁵⁄² = (3 × 25)/2 = 75/2

Now, the equation is: 17⁄3 + 75⁄2 + c = –4

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Step 3: Add the Two Fractions

To combine 17⁄3 and 75⁄2, find the least common denominator (LCD). The LCD of 3 and 2 is 6.

Convert each fraction:

• 17⁄3 = (17 × 2)/6 = 34⁄6
• 75⁄2 = (75 × 3)/6 = 225⁄6

Add them: 34⁄6 + 225⁄6 = (34 + 225)/6 = 259⁄6

Now rewrite the equation: 259⁄6 + c = –4

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Step 4: Express –4 as a Fraction with Denominator 6

-4 = –4/1 = –24⁄6

Now equation becomes: 259⁄6 + c = –24⁄6

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Step 5: Isolate c

Subtract 259⁄6 from both sides: c = –24⁄6 – 259⁄6 = (–24 – 259)/6 = –283⁄6

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Final Result

The solution to the equation 7 – ⁴⁄³ + 3·(²⁵⁄²) + c = –4 is: c = –283⁄6

This confirms the original equation holds true: 7 – ⁴⁄³ + 3·(²⁵⁄²) – 283⁄6 = –4

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Why This Matters – Applications of Linear Equations

Understanding how to manipulate and simplify such equations is crucial in:

• Physics: modeling motion with varying rates
• Economics: balancing budgets with variable costs
• Engineering: solving systems under constraints
• Computer Science: addressing algorithmic complexity

Mastering step-by-step simplification builds a strong foundation for advanced problem solving.

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Summary

To solve equations like 7 – ⁴⁄³ + 3·(²⁵⁄²) + c = –4:

1. Convert all terms to common denominators where possible
2. Perform arithmetic on fractions and mixed numbers
3. Combine constants
4. Isolate the unknown variable c

Mastery of these steps ensures clarity and accuracy in algebra — a key skill across STEM disciplines.

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Further Resources

• Practice combining fractions with different denominators
• Learn to convert between mixed numbers and improper fractions
• Explore solvers for step-by-step algebraic problem practice
• Use real-world word problems to reinforce equation solving

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Tagline: Perfect your algebra—simplify, combine, isolate, and solve with confidence!