Solved problems: algebraic expressions from word phrases (addition & subtraction)

📚 Understanding Algebraic Expressions from Word Phrases

Algebraic expressions are mathematical phrases that combine numbers, variables (letters representing unknown values), and mathematical operations. Translating word phrases into algebraic expressions is a fundamental skill in algebra. It allows us to solve real-world problems by representing them mathematically. This guide focuses on addition and subtraction.

📜 Historical Context

The use of symbols to represent unknown quantities dates back to ancient civilizations. Egyptians and Babylonians used hieroglyphs and cuneiform, respectively, to solve algebraic problems. However, the systematic use of letters as variables and the development of modern algebraic notation emerged during the Islamic Golden Age and later in Europe, particularly with mathematicians like François Viète.

➕ Key Principles for Addition

• 🤝Sum: Represents the result of adding two or more numbers.
• ⬆️Increased by: Indicates adding a value to another.
• ➕Plus: A straightforward addition operation.
• 🚀More than: Adding to a base value.
• ✨Total: The cumulative result of addition.

➖ Key Principles for Subtraction

• 🧮Difference: The result of subtracting one number from another.
• ⬇️Decreased by: Indicates reducing a value.
• ⛔Less: Similar to decreased by.
• 📉Less than: Subtracting from a base value (order matters!).
• 🗑️Minus: Direct subtraction.

📝 Constructing the Expressions

When translating word phrases, identify the keywords that indicate addition or subtraction. Let's look at how to build algebraic expressions:

1. ✅Identify Variables: Assign letters to represent unknown quantities. For instance, let $x$ represent a number.
2. ➕Recognize Addition Phrases: Translate phrases like "a number plus five" into $x + 5$.
3. ➖Recognize Subtraction Phrases: Translate phrases like "seven less than a number" into $x - 7$ (note the order!).

💡 Real-World Examples

Let's convert some word problems into algebraic expressions:

✍️ Practice Quiz

Convert the following word phrases into algebraic expressions:

1. Fourteen plus a number.
2. A number increased by eleven.
3. The sum of a number and six.
4. A number more than twenty.
5. The difference between a number and fifteen.
6. A number decreased by three.
7. Eight less than a number.

Solutions:

1. $14 + x$
2. $x + 11$
3. $x + 6$
4. $20 + x$
5. $x - 15$
6. $x - 3$
7. $x - 8$

🎯 Conclusion

Mastering the translation of word phrases into algebraic expressions is essential for success in algebra and beyond. By understanding the key principles and practicing regularly, you can confidently tackle more complex mathematical problems.