3n + 4p = 7.80 \quad \text{(1)}
Understanding the Linear Equation: 3n + 4p = 7.80
When solving equations like 3n + 4p = 7.80, we dive into the foundational world of algebra, unlocking the relationship between two variables — typically representing cost, quantity, or rate. This equation serves as a powerful model in various real-world contexts such as economics, budgeting, and business pricing.
What Is the Equation 3n + 4p = 7.80?
The equation 3n + 4p = 7.80 is a linear Diophantine expression involving two variables:
- n often represents the quantity of one item, such as a unit of product A, priced at $3 per unit.
- p stands for the quantity of another item, say product B, at $4 per unit.
The right-hand side, 7.80, is the total cost — the sum of costs from buying n units of product A and p units of product B.
In short: Total cost = (price per unit × quantity) for both products = $7.80
Solving for One Variable
To make sense of this equation, solving for one variable in terms of the other clarifies relationships within the system.
Solving for p: Start from equation (1): 3n + 4p = 7.80
Subtract 3n from both sides: 4p = 7.80 – 3n Then divide both sides by 4: p = (7.80 – 3n) / 4
This formula lets you compute the value of p for any given n, as long as the result is a non-negative real number if quantities must be non-negative.
Practical Applications
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Budgeting and Shopping Imagine buying oranges (n) at $3 per kg and apples (p) at $4 per kg. With a fixed budget of $7.80, this equation helps determine how many of each fruit you can buy.
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Cost Analysis in Business In manufacturing, the equation models total production costs based on units of two raw materials or labor tiers.
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Market Equilibrium Studies Social scientists and economists use similar equations to model supply-demand scenarios where two variables influence a known total cost or revenue.
Finding Integer Solutions
For real-life scenarios where only whole units can be purchased, 3n + 4p = 7.80 poses a special challenge — full solutions in integers (wholesale buying, inventory units) rather than decimals.
To find integer solutions accurately:
- Rearrange equation (1): 3n = 7.80 – 4p
- Test small whole-number values of p until 7.80 – 4p is divisible by 3 and produces a non-negative n.
Example: Try p = 1 → 3n = 7.80 – 4(1) = 3.80 → n = 3.80 / 3 ≈ 1.27 (not integer)
Try p = 0.50 → 3n = 7.80 – 4(0.50) = 5.80 → n ≈ 1.93 (still not ideal)
A better approach is to clear decimals: Multiply entire equation 3n + 4p = 7.80 by 10:
30n + 40p = 78
Now work with integers: 15n + 20p = 39
Try small integer values for n and p that satisfy this equation — tools like elimination or substitution make finding exact cost combinations possible.
Conclusion
The equation 3n + 4p = 7.80 is a simple yet effective representation of a real-world linear relationship involving two quantities and their costs. Whether used in budgeting, pricing, or data modeling, understanding how to interpret and solve this equation empowers better decision-making. For practical, whole-number solutions, adjusting or scaling the equation appropriately unlocks greater precision.
Key Takeaways
- 3n + 4p = 7.80 models combined costs of two items.
- Solving for p or n helps explore trade-offs in purchasing or planning.
- The equation supports budgeting, business analysis, and economic modeling.
- Integer solutions require careful testing or scaling to eliminate decimals.
Keywords: 3n + 4p = 7.80, linear equation, algebra, cost calculation, budgeting, cost analysis, integer solutions, household budget, economics equation.
