3n + 4p = 7.80 \quad \text(1) - Decision Point
Understanding the Linear Equation: 3n + 4p = 7.80
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.
Understanding the Context
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.
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Key Insights
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
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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
-
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. -
Cost Analysis in Business
In manufacturing, the equation models total production costs based on units of two raw materials or labor tiers.
- 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.
