Equations, Formulas, and Equivalence Transformations

Fundamentals of Equations and Formulas

• Equation (Gleichung): An equation consists of two algebraic terms joined by an equality sign ($=$).
• Formula (Formel): A formula is an equation containing multiple variables. It describes how variables, physical or mathematical quantities ($\text{Größen}$), and numbers relate to one another.

Setting up Equations and Formulas

• Formulating Algebraic Equations (Problem 556):

  • Variable definitions:
    • $m$: Number of girls ($\text{Anzahl der Mädchen}$)
    • $b$: Number of boys ($\text{Anzahl der Buben}$)
  • Relational equations:
    • Part a: $m + 2 = b$
    • Part b: $2 \cdot (m - 1) = b$

• Travel Bus Pricing Formula (Problem 559):

  • Variable definitions:
    • $b$: Base price for the charter bus ($\text{Preis für Reisebus}$)
    • $f$: Ticket price for each individual passenger ($\text{Preis für jeden Fahrgast}$)
    • $n$: Total number of passengers ($\text{Anzahl der Fahrgäste}$)
    • $P$: Total price
  • 1) General formula for total price $P$:
    • $P = b + n \cdot f$
  • 2) Evaluation given specific parameter values ($b = 50\,\text{€}$, $f = 7\,\text{€}$, $n = 12$):
    • $P = 50 + 12 \cdot 7$
    • $P = 50 + 84$
    • $P = 134\,\text{€}$

Geometric Formulas for Perimeter and Area

• Perimeter and Area Expressions (Problem 564):

  • Part a:
    • 1) Perimeter ($U$): $U = 2 \cdot 4u + 2 \cdot 4v + 2 \cdot 3v = 8u + 8v + 6v = 8u + 14v$
    • 2) Area ($A$): $A = 10uv$
  • Part b:
    • 1) Perimeter ($U$): $U = 20x$
    • 2) Area ($A$): $A = 9x^2$

• Perimeter and Area Expressions (Problem 565):

  • Part a:
    • 1) Perimeter ($U$): $U = 8a$
    • 2) Area ($A$): $A = 3a^2$
  • Part b:
    • 1) Perimeter ($U$): $U = 8x + 2y$
    • 2) Area ($A$): $A = 3x \cdot y - x^2$
  • Part c:
    • 1) Perimeter ($U$): $U = 4u + 4v$
    • 2) Area ($A$): $A = u^2 - v^2$

Equivalent Equations (Äquivalente Gleichungen)

• Definition: Equivalent equations ($\text{Äquivalente Gleichungen}$) are equations that possess the exact same solution set ($\text{mit der gleichen Lösung}$).
• Examples of Equivalent Sets (Problem 568):
  • Part a:
    • Base equation: $5x = 10$
    • Solution: $x = 2$
    • Equivalent equations sharing the solution $x = 2$:
      • $15x = 30$
      • $5x + 3 = 13$
  • Part c:
    • Base equation: $3 - 2x = 1$
    • Solution: $x = 1$
    • Equivalent equations sharing the solution $x = 1$:
      • $-2x = -2$
      • $2x = 2$

Equivalence Transformations (Äquivalenzumformungen)

• Core Principle: Equivalence transformations ($\text{Äquivalenzumformungen}$) convert equations into equivalent equations. The solution of an equation remains unchanged when performing valid algebraic operations on both sides of the equality sign.

• Valid Operations: An equation's solution set does not change if on both sides:

  • The same number is added ($\text{dieselbe Zahl addiert}$).
  • The same number is subtracted ($\text{dieselbe Zahl subtrahiert}$).
  • Multiplied by the same non-zero number ($\text{mit derselben Zahl } \neq 0 \text{ multipliziert}$).
  • Divided by the same non-zero number ($\text{durch dieselbe Zahl } \neq 0 \text{ dividiert}$).
  • The same multiple of the unknown variable is added or subtracted ($\text{dasselbe Vielfache der Unbekannten addiert bzw. subtrahiert}$).

• Step-by-Step Transformation Examples (Problem 572):

  • Part a:
    • Given equation: $x + 6 = 1$
    • Transformation: Subtract $6$ from both sides ($\mid -6$)
    • Resulting solution: $x = -5$
  • Part b:
    • Given equation: $7x = 21$
    • Transformation: Divide both sides by $7$ ($\mid :7$)
    • Resulting solution: $x = 3$
  • Part c:
    • Given equation: $3t = 3t$
    • Transformation: Divide both sides by $3$ ($\mid :3$)
    • Resulting equation: $t = t$
  • Part d:
    • Given equation: $3x - 7 = 2x + 1$
    • Transformation step: Subtract $2x$ from both sides ($\mid -2x$)
    • Resulting step: $x - 7 = 1$
  • Part e:
    • Given equation: $5z + 8 = 1 - 2z$
    • Transformation step: Subtract $1$ from both sides ($\mid -1$)
    • Resulting step: $5z + 7 = -2z$