Solving Equations and Translating Expressions

Solving Equations for Unknown Values

• Keep the equation balanced while isolating the unknown.
• Example:
  • $6(x+4.2) = 36$
  • $6(x+4.2) / 6 = 36 / 6$
  • $x + 4.2 = 6$
  • $x + 4.2 - 4.2 = 6 - 4.2$
  • $x = 1.8$
• Example:
  • $6(x+3)=3(x+10)$
  • $6x + 18 = 3x + 30$
  • $6x + 18 - 3x = 3x + 30 - 3x$
  • $3x+18=30$
  • $3x+18 - 18 = 30 - 18$
  • $3x = 12$
  • $3x / 3 = 12 / 3$
  • $x = 4$

Types of Solutions for Linear Equations

• No solution: Transforms to $a = b$, where $a$ and $b$ are different numbers.
• One solution: Transforms to $x = a$, where $x$ is a variable and $a$ is a number.
• Infinitely many solutions: Transforms to $a = a$, where $a$ is a number.
• Example:
  • $2 + 4x + 20 = 6x + 22 - 2x$
  • $4x + 22 = 6x + 22 - 2x$
  • $4x + 22 = 4x + 22$
  • $4x + 22 - 4x = 4x + 22 - 4x$
  • $22 = 22$
  • The equation has infinitely many solutions.

Translating Words into Mathematical Expressions

• Recognize which operations to use:
  • Addition: more than, increase, plus, sum
  • Subtraction: less than, decrease, minus, difference
  • Multiplication: times, product
  • Division: quotient
• Ensure correct order of operations:
  • Parentheses
  • Exponents
  • Multiplication and division (left to right)
  • Addition and subtraction (left to right)
• Represent values: Use a variable or symbol for unknown values.
• Example:
  • "decrease 13 by the product of 4 and some number" translates to $13 - 4x$
• Example:
  • "6 times the sum of 12 and some number"