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.

3n + 4p = 7.80 \quad \text{(1)}

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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\(d = 10 \times 4 + \frac{1}{2} \times 10 \times 4^2\)
\(d = 40 + 80 = 120\) meters.
The ball will travel 120 meters in 4 seconds.

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Final Thoughts

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

  1. 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.

  2. Cost Analysis in Business
    In manufacturing, the equation models total production costs based on units of two raw materials or labor tiers.

  1. 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.