Solved Problems: One-Step Equations with Addition (X + A = B)

📚 What is a One-Step Equation with Addition?

A one-step equation with addition is an algebraic equation that can be solved in just one step using the inverse operation of addition: subtraction. The general form is $x + a = b$, where $x$ is the variable we want to find, and $a$ and $b$ are constants (numbers).

📜 History and Background

The concept of algebraic equations dates back to ancient civilizations. Egyptians and Babylonians used symbols to represent unknown quantities and developed methods to solve for them. Diophantus, a Greek mathematician, is often called the "father of algebra" for his work in developing symbolic notation and solving algebraic problems. Solving equations has evolved significantly over time, but the fundamental principle of isolating the variable remains the same.

📌 Key Principles for Solving $x + a = b$

• ⚖️ The Golden Rule: What you do to one side of the equation, you MUST do to the other side to maintain equality.
• ➖ Inverse Operation: Use subtraction, the inverse of addition, to isolate the variable $x$.
• 🎯 Goal: Isolate $x$ on one side of the equation. This means getting $x$ by itself.
• 🔢 Apply Subtraction: Subtract $a$ from both sides: $x + a - a = b - a$. This simplifies to $x = b - a$.

💡 Step-by-Step Solution

1. 📝 Identify the equation: Recognize the form $x + a = b$.
2. ➖ Subtract 'a' from both sides: This cancels out the $+a$ on the left.
3. ➕ Simplify: Calculate the result on the right side ($b - a$).
4. ✅ Write the solution: The solution is $x = b - a$.

🌍 Real-World Examples

Example 1:

Suppose you have a bag of marbles. You know that if you add 5 more marbles to the bag, you'll have a total of 12 marbles. How many marbles were originally in the bag?

Equation: $x + 5 = 12$

Solution: Subtract 5 from both sides: $x + 5 - 5 = 12 - 5$, so $x = 7$.

Example 2:

You are baking cookies and need 24 cookies for a party. You already have 8 cookies. How many more cookies do you need to bake?

Equation: $x + 8 = 24$

Solution: Subtract 8 from both sides: $x + 8 - 8 = 24 - 8$, so $x = 16$.

Example 3:

John has a certain number of apples. His friend gives him 7 more apples, and now John has 15 apples. How many apples did John have originally?

Equation: $x + 7 = 15$

Solution: Subtract 7 from both sides: $x + 7 - 7 = 15 - 7$, so $x = 8$.

📝 Practice Quiz

• ❓ Solve for x: $x + 3 = 9$
• ❓ Solve for x: $x + 11 = 20$
• ❓ Solve for x: $x + 6 = 14$
• ❓ Solve for x: $x + 2 = 10$
• ❓ Solve for x: $x + 4 = 16$
• ❓ Solve for x: $x + 9 = 17$
• ❓ Solve for x: $x + 5 = 13$

🔑 Answers to the Quiz

• ✅ $x = 6$
• ✅ $x = 9$
• ✅ $x = 8$
• ✅ $x = 8$
• ✅ $x = 12$
• ✅ $x = 8$
• ✅ $x = 8$

🧠 Conclusion

One-step equations with addition are the foundation for solving more complex algebraic problems. By understanding the principle of inverse operations and practicing regularly, you can master these equations with ease. Remember, math is a journey, not a destination! Keep practicing, and you'll become a math whiz in no time! 🚀