Solving 3z – 8z = 2 + 5: A Step-by-Step Guide to Solve Linear Equations

Mathematics often presents challenges in the form of equations that demand step-by-step logic to unlock their solutions. One such equation that many students encounter is 3z – 8z = 2 + 5. While it might seem simple at first glance, understanding how to solve it properly not only helps with algebra basics but builds a strong foundation for more complex problem-solving.

In this article, we’ll break down how to solve 3z – 8z = 2 + 5, explain the important algebraic steps, and highlight why mastering such equations is key in both school and real-world applications.

Understanding the Context

What Is the Equation: 3z – 8z = 2 + 5?

At first glance, this is a linear equation involving one variable, z. Though the expression looks short, correctly simplifying and solving it requires attention to order of operations and basic algebraic principles.

Let’s rewrite it clearly:

3z - 8z = 2 + 5

Key Insights

3z – 8z = 2 + 5

Step 1: Simplify Both Sides

Start by simplifying both sides using arithmetic rules:

  • Left side: 3z – 8z = -5z
  • Right side: 2 + 5 = 7

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Not always divisible by higher powers (e.g., $ n = 1 $: product = 24, not divisible by 48).
Thus, the largest integer that must divide the product of any four consecutive integers is $ \boxed{24} $.
Question: A mammalogist observes that a species of Arctic foxes returns to its den every 5 days, and another species every 7 days. If both species were seen at the den on day 1, on which day will they next be seen together?

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Final Thoughts

Now the equation becomes:

-5z = 7

Step 2: Isolate the Variable z

To solve for z, divide both sides by –5:

z = 7 ÷ (–5)
z = –7/5

This fraction can also be written as the decimal –1.4, offering two common ways to express the solution.

Why Is Solving Such Equations Important?

Solving linear equations like 3z – 8z = 2 + 5 is fundamental not only in algebra but in many practical scenarios: