Solving Equations Worksheet Notes

Key Concepts

• The worksheet is focused on solving linear equations to find the value of variables, primarily x.
• The equations range from simple one-variable equations to more complex forms involving multiple steps.

Types of Problems

1. Simple Linear Equations

• Format: x + a = b
  • Examples:
    • a) x + 3 = 9
    • Solve by subtracting 3 from both sides.
  • Solution: Subtract the constant from both sides to isolate x.

2. Multiplicative Equations

• Format: ax = b
  • Examples:
    • a) 2x = 18
    • Solve by dividing both sides by the coefficient of x.
  • Solution: Divide both sides by the coefficient to solve for x.

3. Two-Step Equations

• Format: ax + b = c
  • Examples:
    • a) 3x + 1 = 13
    • Solve by first subtracting the constant, then dividing by the coefficient.
  • Solution: Perform inverse operations in steps to isolate x.

4. Equations with Variables on Both Sides

• Format: ax + b = cx + d
  • Examples:
    • a) 4x + 1 = 2x + 7
    • Reorganize to get all terms with x on one side.
  • Solution: Use addition or subtraction to get all x terms on one side and constants on the other, then solve.

5. Word Problems

• Scenario: Relating a real-world situation in an equation.
  • Problem example involves marbles:
    • Equation: 3x = x + 24
  • Solution: Solve for x which represents the number of marbles in a bag.

Solutions Overview

• Basic Linear: Subtract or add to isolate x.
• Multiplicative: Divide by coefficient.
• Two-Step: Inverse operations sequentially.
• Variable on Both Sides: Collect like terms and solve.
• Word Problems: Translate words into equations, solve for x.

Answer Key

• Section 1 answers range from x = -11 to positive values.
• Section 2 involves different variables (t, d, f) with solutions provided.
• Section 3 involves two-step equations with solutions such as x = 4.
• Section 4 involves solving for x with operations on both sides, e.g., x = 3.
• Section 5 and 6 provide real-world context and solutions such as x = 12 for marbles in a bag.