2️⃣ Mastering Equations for Better Decision-Making — Slides

Mathematical Fundamentals · MS01-001-G

Learning Objectives

• Simplify and rewrite algebraic expressions
• Reveal hidden structure using a common factor or a notable identity
• Solve first-degree equations in a business or economic context

1. Expanding

Distributivity: multiply a factor across a sum or difference.

\[a(b+c) = ab + ac\]

Example:

\[3(2x - 5) = 6x - 15\]

Expanding removes parentheses; use it to simplify complex expressions.

2. Factoring

Extracting a common factor: the reverse of expanding.

\[ab + ac = a(b + c)\]

Example:

\[6x^2 + 9x = 3x(2x + 3)\]

Factoring reveals hidden structure — always look for the greatest common factor first.

3. Notable Identities

Three Essential Identities

\[( a + b )^2 = a^2 + 2ab + b^2\]

\[( a - b )^2 = a^2 - 2ab + b^2\]

\[(a + b)(a - b) = a^2 - b^2\]

Common Mistake

\[(a + b)^2 \neq a^2 + b^2\]

Do not forget the middle term \(2ab\).

4. First-Degree Equations

Rule: isolate the unknown by applying the same operation to both sides.

General form:

\[ax + b = c \Rightarrow x = \frac{c - b}{a}\]

Example:

\[5x - 8 = 12\] \[5x = 20\] \[x = 4\]

Worked Example — Break-Even Point

A company has:

• Fixed costs: €10,000
• Variable cost: €8 per unit
• Revenue: €13 per unit

Set revenue equal to total cost and solve:

\[13x = 10{,}000 + 8x\] \[13x - 8x = 10{,}000\] \[5x = 10{,}000\] \[x = 2{,}000 \text{ units}\]

The company must sell 2,000 units to break even.

Summary

1. Expanding and factoring are inverse operations — expanding removes parentheses, factoring reintroduces them to reveal structure.
2. Notable identities are shortcuts — memorise the three forms: \((a+b)^2\), \((a-b)^2\), and \((a+b)(a-b)\).
3. To solve a first-degree equation, isolate the unknown by applying inverse operations to both sides.

📖 Full Reference

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

→ Module 2: Mastering Equations