Understanding Two-Step Equations

Understanding Two-Step Equations

• Two-step equations are algebraic equations that require two steps to solve for the variable.
• They involve a combination of addition/subtraction and multiplication/division operations.
• The goal is to isolate the variable on one side of the equation.

Identifying Two-Step Equations

• An equation is a two-step equation if it contains a variable, and at least two operations are performed on the variable to isolate it.

Solving Two-Step Equations

• Step 1: Undo addition or subtraction: Add or subtract the constant term from both sides of the equation to isolate the term with the variable.
• Step 2: Undo multiplication or division: Divide or multiply both sides of the equation by the coefficient of the variable.
• Example: Solve 2x + 5 = 11
  • Subtract 5 from both sides: 2x = 6
  • Divide both sides by 2: x = 3

Key Concepts

• Inverse Operations: The operations used to isolate the variable are inverse operations—they "undo" each other.
• Properties of Equality: The following properties are crucial for solving equations:
  • Addition Property of Equality: If a = b, then a + c = b + c.
  • Subtraction Property of Equality: If a = b, then a - c = b – c.
  • Multiplication Property of Equality: If a = b, then a * c = b * c.
  • Division Property of Equality: If a = b, then a / c = b / c (assuming c ≠ 0).
• Checking Solutions: Substitute the solution into the original equation to verify it is correct.

Common Mistakes

• Forgetting the order of operations.
• Incorrect applying inverse operations.
• Making errors with signs, especially when subtracting negative numbers.
• Dividing or multiplying both sides by zero, which is an invalid operation.

Examples of Different Types of Two-Step Equations

• Subtraction and multiplication: 3x - 7 = 8
  • Add 7 to both sides: 3x = 15
  • Divide both sides by 3: x = 5.
• Addition and division: (y/4) + 2 = 6
  • Subtract 2 from both sides: (y/4) = 4
  • Multiply both sides by 4: y = 16.
• Negative Coefficients: -2x + 9 = 1
  • Subtract 9 from both sides: −2x = −8
  • Divide both sides by (−2): x = 4.
• Fractions: (x/3) + 4 = 7
  • Subtract 4 from both sides: (x/3) = 3.
  • Multiply both sides by 3: x = 9.

Word Problems Involving Two-Step Equations

• Translate word problems into algebraic equations.
• Identify the unknown quantities and assign variables.
• Solve the two-step equation to find the unknown values.

Practice Problems:

• Solve 5x − 2 = 13
• Solve 11 + 2y/3 = 7
• A pizza costs $15. If this includes the cost of additional toppings ($4), what is the base price of the pizza? Equation: x + 4 = 15. Solution: x = $11.