Constructing and Solving Equations

2.5 Constructing and Solving Equations

Overview of Key Terms

• Formula: A rule that shows the relationship between two or more quantities (variables). It must contain an equals sign.
• Expression: A statement involving one or more variables, but without an equals sign.
• Equation: An expression that contains an unknown number, must have an equals sign, and can be solved to find the value of the unknown number.

Definitions and Examples

• Formula Examples:
    - $F = ma$ (Relation between force, mass, and acceleration)
    - $v = u + 10t$ (Relation between initial velocity, final velocity, and time)

• Expression Examples:
    - $3x - 7$ (Involves variable x, no equals sign)
    - $a^2 + 2b$ (Involves variables a and b, no equals sign)
  

• Equation Examples:
    - $3x - 7 = 14$ (Contains unknown number x, can be solved for x)
    - $4 = 6y + 22$ (Contains unknown number y, can be solved for y)

Constructing Equations

When faced with a problem, constructing or writing an equation is often necessary to find a solution.

Worked Example 2.5

Part a: Classifying Mathematical Statements

Determine whether the following are a formula, an expression, or an equation:

1. i. $4c + 3e$
      - Classification: Expression
      - Explanation: Involves two variables but lacks an equals sign.

2. ii. $P = 8h + b$
      - Classification: Formula
      - Explanation: Represents a rule showing the relationship between three quantities: P, h, and b.

3. iii. $9k - 2 = 16$
      - Classification: Equation
      - Explanation: Contains an unknown number k, has an equals sign, and can be solved for k.

Part b: Geometry Problem

• A diagram shows a rectangle with the following dimensions given:
    - Width: $3(x + 3)$ cm
    - Height: $5y - 4$ cm
    - Perimeter: $24$ cm

Key Terms

• Construct: The action of writing or creating an equation or expression based on provided information or relationships.