Solving one-step equations using inverse operations: a complete guide

📚 Understanding One-Step Equations

One-step equations are algebraic equations that can be solved in just one step. They involve a variable (usually represented by a letter like $x$ or $y$), a constant, and an operation (addition, subtraction, multiplication, or division). The goal is to isolate the variable on one side of the equation to find its value.

🕰️ History and Background

The concept of solving equations dates back to ancient civilizations. Egyptians and Babylonians used methods to solve linear equations. However, the modern algebraic notation we use today developed gradually over centuries. The use of symbols like '$x$' and '$=$' became standardized in the 16th and 17th centuries, making algebraic manipulations more accessible.

🔑 Key Principles: Inverse Operations

Inverse operations are operations that "undo" each other. Using inverse operations is the key to solving one-step equations.

• ➕ Addition and Subtraction: These are inverse operations. If an equation involves adding a number to the variable, subtract that number from both sides of the equation to isolate the variable. Conversely, if the equation involves subtracting a number, add that number to both sides.
• ➗ Multiplication and Division: These are also inverse operations. If an equation involves multiplying the variable by a number, divide both sides of the equation by that number. If the equation involves dividing the variable by a number, multiply both sides by that number.
• ⚖️ Maintaining Balance: The golden rule of solving equations is to maintain balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the equation true.

✏️ Solving One-Step Equations: Examples

Let's look at some examples to illustrate how to solve one-step equations using inverse operations:

1. Example 1: Addition

   Solve for $x$: $x + 5 = 12$

   To isolate $x$, subtract 5 from both sides:

   $x + 5 - 5 = 12 - 5$

   $x = 7$

2. Example 2: Subtraction

   Solve for $y$: $y - 3 = 8$

   To isolate $y$, add 3 to both sides:

   $y - 3 + 3 = 8 + 3$

   $y = 11$

3. Example 3: Multiplication

   Solve for $a$: $3a = 15$

   To isolate $a$, divide both sides by 3:

   $\frac{3a}{3} = \frac{15}{3}$

   $a = 5$

4. Example 4: Division

   Solve for $b$: $\frac{b}{4} = 6$

   To isolate $b$, multiply both sides by 4:

   $4 \cdot \frac{b}{4} = 6 \cdot 4$

   $b = 24$

💡 Real-World Applications

• Budgeting: Calculating expenses or income.
• Cooking: Adjusting recipe quantities.
• Travel: Determining distances or travel times.

✔️ Conclusion

Solving one-step equations is a fundamental skill in algebra. By understanding and applying inverse operations, you can easily isolate variables and find their values. This skill is essential for more advanced mathematical concepts and has numerous real-world applications.