Understanding the Equation: Solving 0.05x + 0.08(10,000 - x) = 680

When faced with a linear equation like 0.05x + 0.08(10,000 - x) = 680, it might seem daunting at first—but breaking it down step by step makes solving it straightforward. Whether you're a student learning algebra or a professional working through real-world financial calculations, understanding how to solve equations systematically is essential.

Understanding the Context

What is the Equation?

The equation 0.05x + 0.08(10,000 - x) = 680 models a weighted average scenario, commonly used in finance, cost analysis, or profit modeling. It balances two different rates applied to parts of a total quantity.

In this case:
- 0.05x represents 5% of a quantity x
- 0.08(10,000 - x) represents 8% applied to the remaining (10,000 - x)

Their sum equals 680, likely modeling a total weighted cost or return.

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Key Insights

Step 1: Expand the Expression

Start by distributing 0.08 across the parenthesis:

\[
0.05x + 0.08 \ imes 10,000 - 0.08x = 680
\]

Calculate \(0.08 \ imes 10,000 = 800\), so the equation becomes:

Continue Reading

\frac{x^2}{16} - \frac{y^2}{9} = 1.
This equation is in the standard form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) is the center. Here, \(h = 0\) and \(k = 0\), so the center of the hyperbola is at the origin.
\(\boxed{(0, 0)}\)

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Final Thoughts

\[
0.05x + 800 - 0.08x = 680
\]

Step 2: Combine Like Terms

Combine the x terms:

\[
(0.05 - 0.08)x + 800 = 680 \
-0.03x + 800 = 680
\]

Step 3: Isolate the Variable

Subtract 800 from both sides:

\[
-0.03x = 680 - 800 \
-0.03x = -120
\]

Now divide both sides by \(-0.03\):