Understanding Two-Step Equations

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Questions and Answers

Which of the following is NOT a two-step equation?

  • x + 2 = 5 (correct)
  • 3y - 4 = 10
  • 2x + 5 = 11
  • 4z - 1 = 7

Solve for x: 5x - 3 = 12.

  • x = 7
  • x = 5
  • x = 2
  • x = 3 (correct)

What is the first step in solving the equation (y/3) + 5 = 8?

  • Add 5 to both sides
  • Multiply both sides by 3
  • Divide both sides by 3
  • Subtract 5 from both sides (correct)

Which of the following is the correct solution to the equation -2x + 7 = 1?

<p>x = 4 (A)</p> Signup and view all the answers

What is the correct order of operations for solving a two-step equation?

<p>Addition/Subtraction, then Multiplication/Division (C)</p> Signup and view all the answers

A store sells apples for $1.50 each and oranges for $0.75 each. Sarah bought 3 apples and some oranges, spending a total of $7.50. How many oranges did she buy?

<p>4 (B)</p> Signup and view all the answers

Which of the following is NOT a step in solving the equation 3x - 5 = 10?

<p>Subtract 10 from both sides (A)</p> Signup and view all the answers

What is the inverse operation of multiplying by 4?

<p>Dividing by 4 (A)</p> Signup and view all the answers

Flashcards

Two-Step Equations

Algebraic equations needing two steps to solve for a variable.

Inverse Operations

Operations that undo each other, aiding in isolating variables.

Solving Steps

The process of first undoing addition/subtraction, then multiplication/division to isolate a variable.

Addition Property of Equality

If a = b, then a + c = b + c maintains equality.

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Subtraction Property of Equality

If a = b, then a - c = b - c keeps the equation valid.

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Common Mistakes

Errors often made include incorrect order of operations and misapplying inverse operations.

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Checking Solutions

Substituting back into the original equation to verify correctness of the solution.

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Word Problems

Translate textual problems into two-step algebraic equations to find unknown values.

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Study Notes

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.

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