Two Step Equations That Equal 1

Two-Step Equations That Equal 1: A Comprehensive Guide

Solving two-step equations is a fundamental skill in algebra. While the process remains consistent regardless of the solution, focusing on equations that equal 1 provides a unique opportunity to practice manipulating equations and understanding the underlying principles. This comprehensive guide will delve into various aspects of solving two-step equations that result in x = 1, covering different equation types, step-by-step solutions, common mistakes to avoid, and advanced applications.

Understanding Two-Step Equations

A two-step equation is an algebraic equation that requires two steps to isolate the variable and solve for its value. These equations typically involve addition, subtraction, multiplication, and division operations. The general form of a two-step equation is:

ax + b = c

Where:

• x represents the unknown variable.
• a and b are constants (numbers).
• c is the constant on the other side of the equals sign.

In our case, we're focusing on equations where the solution is x = 1. This means that after completing the two steps, the value of x will always be 1.

Step-by-Step Solution Process

The core principle behind solving two-step equations is to isolate the variable (x) by performing inverse operations. This involves reversing the order of operations (PEMDAS/BODMAS) in reverse.

1. Undo Addition or Subtraction: The first step involves eliminating the constant term (b) added to or subtracted from the term containing the variable (ax). This is achieved by performing the inverse operation on both sides of the equation. If 'b' is added, subtract 'b' from both sides. If 'b' is subtracted, add 'b' to both sides.

2. Undo Multiplication or Division: Once the variable term (ax) is isolated, eliminate the coefficient 'a' by performing the inverse operation. If 'a' is multiplying x, divide both sides by 'a'. If 'a' is dividing x, multiply both sides by 'a'.

Important Note: Remember to perform the same operation on both sides of the equation to maintain balance and ensure the equation remains valid.

Examples of Two-Step Equations Equaling 1

Let's illustrate the process with various examples:

Example 1:

3x + 2 = 5

1. Subtract 2 from both sides: 3x + 2 - 2 = 5 - 2 => 3x = 3
2. Divide both sides by 3: 3x / 3 = 3 / 3 => x = 1

Example 2:

5x - 7 = -2

1. Add 7 to both sides: 5x - 7 + 7 = -2 + 7 => 5x = 5
2. Divide both sides by 5: 5x / 5 = 5 / 5 => x = 1

Example 3:

x/2 + 3 = 4

1. Subtract 3 from both sides: x/2 + 3 - 3 = 4 - 3 => x/2 = 1
2. Multiply both sides by 2: (x/2) * 2 = 1 * 2 => x = 1

Example 4:

-2x + 10 = 8

1. Subtract 10 from both sides: -2x + 10 - 10 = 8 - 10 => -2x = -2
2. Divide both sides by -2: -2x / -2 = -2 / -2 => x = 1

Example 5 (with fractions):

(2/3)x + 1/3 = 1

1. Subtract 1/3 from both sides: (2/3)x + 1/3 - 1/3 = 1 - 1/3 => (2/3)x = 2/3
2. Multiply both sides by 3/2: (3/2) * (2/3)x = (3/2) * (2/3) => x = 1

These examples demonstrate the consistent application of the two-step process. Regardless of the initial coefficients and constants, the manipulation of the equation always leads to x = 1.

Common Mistakes to Avoid

While the process is straightforward, several common errors can lead to incorrect solutions:

• Incorrect Order of Operations: Failing to follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions can lead to errors. Always address addition/subtraction before multiplication/division when isolating the variable.
• Errors in Arithmetic: Simple calculation mistakes are common. Double-check your arithmetic at each step to minimize errors.
• Forgetting to Perform Operations on Both Sides: This is a critical error. Always perform the same operation on both sides of the equation to maintain balance.
• Incorrect Sign Manipulation: Mismanaging negative signs can lead to wrong answers. Pay close attention to the signs of the coefficients and constants.
• Fractional Arithmetic Errors: Working with fractions requires careful attention to detail. Make sure to correctly perform addition, subtraction, multiplication, and division of fractions.

Advanced Applications and Extensions

The concept of two-step equations extends far beyond basic algebraic manipulations. These equations form the foundation for more complex problem-solving in various fields:

• Physics: Many physics problems involve solving equations for unknown variables such as velocity, acceleration, or time.
• Engineering: Engineers frequently use equations to model and analyze systems, requiring the ability to solve for unknown parameters.
• Chemistry: Stoichiometry problems in chemistry often involve solving equations to determine the quantities of reactants and products in chemical reactions.
• Economics: Economic models utilize equations to analyze relationships between variables like supply, demand, and price.
• Computer Science: Algorithmic processes often involve solving equations to control program flow and calculate values.

Solving Real-World Problems with Two-Step Equations

Let's consider a real-world scenario:

Problem: You are saving money to buy a new video game costing $60. You already have $10 saved. If you save $5 each week, how many weeks will it take to save enough money for the game?

Solution:

Let x represent the number of weeks. The equation becomes:

10 + 5x = 60

1. Subtract 10 from both sides: 5x = 50
2. Divide both sides by 5: x = 10

It will take 10 weeks to save enough money for the video game.

Conclusion

Mastering two-step equations is a crucial step in developing strong algebraic skills. Focusing on equations that equal 1 provides valuable practice in manipulating equations and developing a deeper understanding of the underlying principles. By understanding the steps, avoiding common mistakes, and applying these skills to real-world problems, you'll build a strong foundation for tackling more complex mathematical challenges. Remember to practice regularly and seek help when needed to solidify your understanding and excel in your mathematical journey. The more you practice, the more comfortable and confident you will become in solving these types of equations.