How to use inverse operations to solve one-step equations (Grade 7 steps)

📚 Understanding Inverse Operations

In mathematics, an inverse operation is an operation that undoes another operation. Think of it as the opposite action. For example, addition and subtraction are inverse operations, as are multiplication and division. Understanding this concept is crucial for solving equations.

📜 A Brief History

The concept of inverse operations has been around for centuries, evolving alongside algebra itself. Early mathematicians recognized the need to 'undo' operations to find unknown values, leading to the formalization of inverse operations as we know them today. The development of symbolic algebra in the 16th and 17th centuries greatly enhanced the use and understanding of these operations.

🔑 Key Principles of Inverse Operations

• ➕ Addition and Subtraction: Addition is the inverse of subtraction, and vice versa. If you have an equation like $x + 5 = 10$, you subtract 5 from both sides to isolate $x$.
• ➖ Subtraction and Addition: Subtraction is the inverse of addition. If you have an equation like $x - 3 = 7$, you add 3 to both sides to isolate $x$.
• ✖️ Multiplication and Division: Multiplication is the inverse of division, and vice versa. If you have an equation like $3x = 12$, you divide both sides by 3 to isolate $x$.
• ➗ Division and Multiplication: Division is the inverse of multiplication. If you have an equation like $\frac{x}{4} = 6$, you multiply both sides by 4 to isolate $x$.
• ⚖️ Maintaining Balance: When solving equations, always remember to perform the same operation on both sides to maintain equality. This ensures that the equation remains balanced.

🧮 Solving One-Step Equations: A Practical Guide

To solve one-step equations using inverse operations, follow these steps:

1. Identify the Operation: Determine what operation is being applied to the variable (addition, subtraction, multiplication, or division).
2. Apply the Inverse Operation: Perform the inverse operation on both sides of the equation to isolate the variable.
3. Simplify: Simplify both sides of the equation to find the value of the variable.
4. Check Your Solution: Substitute the value you found back into the original equation to verify that it is correct.

➕ Real-World Examples

Let's look at some examples to illustrate how inverse operations are used in solving one-step equations:

1. Example 1: Addition

   Solve for $x$: $x + 7 = 15$

   To isolate $x$, subtract 7 from both sides:

   $x + 7 - 7 = 15 - 7$

   $x = 8$

2. Example 2: Subtraction

   Solve for $y$: $y - 4 = 9$

   To isolate $y$, add 4 to both sides:

   $y - 4 + 4 = 9 + 4$

   $y = 13$
3. Example 3: Multiplication

   Solve for $z$: $5z = 25$

   To isolate $z$, divide both sides by 5:

   $\frac{5z}{5} = \frac{25}{5}$

   $z = 5$

4. Example 4: Division

   Solve for $a$: $\frac{a}{3} = 6$

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

   $3 \cdot \frac{a}{3} = 6 \cdot 3$

   $a = 18$

📝 Practice Quiz

Solve the following one-step equations:

1. $x + 3 = 8$
2. $y - 5 = 2$
3. $2z = 10$
4. $\frac{a}{4} = 5$
5. $b + 6 = 11$
6. $c - 2 = 7$
7. $6d = 24$

💡 Solutions to Practice Quiz

1. $x = 5$
2. $y = 7$
3. $z = 5$
4. $a = 20$
5. $b = 5$
6. $c = 9$
7. $d = 4$

🎯 Conclusion

Mastering inverse operations is a fundamental skill in algebra. By understanding how to 'undo' mathematical operations, you can confidently solve one-step equations and build a strong foundation for more complex algebraic problems. Keep practicing, and you'll become a math whiz in no time!