0.05x + 0.08(10,000 - x) = 680 - Decision Point
Understanding the Equation: Solving 0.05x + 0.08(10,000 - x) = 680
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:
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\[
0.05x + 800 - 0.08x = 680
\]
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 \(-0.03\):
