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.

Understanding the Context

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
7\left(-\frac{4}{3}\right) + 3\left(\frac{25}{2}\right) + c = -4 \Rightarrow -\frac{28}{3} + \frac{75}{2} + c = -4

Key Insights

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

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Final Thoughts

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

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