Understanding the Equation: 3z – 5 = 8z + 2 – A Step-by-Step Algebra Breakdown
Solving linear equations is a foundational skill in algebra, and mastering an equation like 3z – 5 = 8z + 2 opens the door to success in many math challenges. Whether you’re a student learning algebra or someone brushing up on basics, solving for z offers clear, logical steps that build confidence and problem-solving strength. In this article, we’ll walk through solving the equation 3z – 5 = 8z + 2 step-by-step, explain key algebraic principles, and highlight why this type of problem matters.
What Does the Equation Mean?
The equation 3z – 5 = 8z + 2 represents a balance between two expressions involving the variable z. The left side (3z – 5) and the right side (8z + 2) are equal, and our goal is to isolate z to find its value. Understanding this balance is crucial — algebra isn’t magic, but logic in action.
Step-By-Step Solution
1. Isolate z terms on one side
Begin by moving all terms with z to one side of the equation. Subtract 3z from both sides:
3z – 5 = 8z + 2
→ –5 = 5z + 2
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#### 2. **Move constant terms to the other side**
Now eliminate the constant **2** by subtracting **2** from both sides:
–5 – 2 = 5z To isolate z, divide both sides by 5:
→ –7 = 5z
3. Solve for z
z = –7 ÷ 5
→ z = –1.4
→ z = – rac{7}{5}
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### Final Answer<br/>
The solution to the equation <strong>3z – 5 = 8z + 2</strong> is:<br/>
<strong>z = – rac{7}{5}</strong> (or –1.4 in decimal form).
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### Why Is This Equation Important?<br/>
Understanding how to solve equations like 3z – 5 = 8z + 2 strengthens foundational algebra skills. These principles apply to:<br/>
- Solving for unknown values in science and engineering problems<br/>
- Budgeting and financial modeling in real life<br/>
- Preparing for advanced math courses such as calculus and linear algebra
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### Tips for Mastering Linear Equations<br/>
- Always balance both sides when moving terms.<br/>
- Keep terms organized — write one side as “z terms” and the other as “constants.”<br/>
- Practice with positive, negative, and fractional coefficients.<br/>
- Use inverse operations consistently.
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### Conclusion<br/>
Solving <strong>3z – 5 = 8z + 2</strong> isn’t just about finding a number; it’s about learning to think clearly, logically, and step by step. With practice, you’ll develop the precision and confidence to tackle more complex equations. Keep practicing, stay curious, and remember: algebra is your tool for unlocking logical solutions!
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