7\left(-\frac43\right) + 3\left(\frac252\right) + c = -4 \Rightarrow -\frac283 + \frac752 + c = -4 - Decision Point
Solving the Equation:
7 ⁻⁴⁄₃ + 3·(²⁵⁄²) + c = –4
▶ Interpreting and Solving Linear Equation
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
Image Gallery
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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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
