3️⃣ Curve Analysis in Economics and Management — Slides

Mathematical Fundamentals · MS01-001-G

Learning Objectives

  • Read and extract information from a graph in the Cartesian plane
  • Find and interpret the equation of a line
  • Solve first-degree systems of two equations
  • Model and solve first-degree inequalities

1. Reading a Graph

Key features to identify when analysing a curve:

Feature How to read it
Intercepts Where the curve crosses the axes
Sign Regions where the curve is positive or negative
Variation Where the curve is increasing or decreasing
Extrema Local maximum or minimum points

2. Equation of a Line

Slope-Intercept Form

\[y = ax + b\]

  • Slope \(a = \dfrac{y_2 - y_1}{x_2 - x_1}\) — rate of change between two points
  • Intercept \(b\) — value of \(y\) when \(x = 0\)

Interpretation of slope:

  • \(a > 0\) → increasing line
  • \(a < 0\) → decreasing line
  • \(a = 0\) → horizontal line

3. Systems of Equations

Two lines intersect at the point that satisfies both equations simultaneously.

Method 1 — Substitution

Express one variable in terms of the other, then substitute into the second equation.

Method 2 — Elimination

Add or subtract the equations to cancel one variable.

General form:

\[y = a_1 x + b_1 \quad \text{and} \quad y = a_2 x + b_2\]

The intersection \((x^*, y^*)\) is the solution of the system.

4. Inequalities

A first-degree inequality of the form \(ax + b > 0\) has an interval as its solution set.

Sign Reversal Rule

When you multiply or divide both sides by a negative number, the inequality sign reverses:

\[-2x > 4 \Rightarrow x < -2\]

Forgetting this rule is one of the most common errors.

Worked Example — Two Pricing Models

  • Model A: total cost \(C_A = 200 + 5x\)
  • Model B: total cost \(C_B = 100 + 8x\)

Find the break-even quantity — set the two costs equal:

\[200 + 5x = 100 + 8x\] \[100 = 3x\] \[x \approx 33.3 \text{ units}\]

  • Below 33 units → Model B is cheaper
  • Above 33 units → Model A is cheaper

Summary

  1. A line is fully described by its slope and intercept — slope gives the rate of change, intercept gives the starting value.
  2. The intersection of two lines solves a two-variable system — use substitution or elimination to find it.
  3. An inequality solution is an interval — remember to reverse the inequality sign when multiplying or dividing by a negative number.

📖 Full Reference

For complete theory, exercises, and worked problems, see the full module:

→ Module 3: Curve Analysis