Welcome to the world of Algebra! Algebra is a fascinating branch of mathematics that has been around for thousands of years. Its origins can be traced back to ancient civilizations, such as Egypt, where it was used to solve practical problems. The ancient Egyptians used the word ‘aha’, meaning ‘heap’, to represent an unknown number. Today, we use letters, like x, to stand for unknown values in equations.
One of the earliest known examples of algebra is found in the Ahmes Papyrus from around 1650 BC. This ancient document contains mathematical problems that require algebraic thinking to solve. Historians believe these problems were used as exercises to train young mathematicians. These skills were likely essential for tasks like building the pyramids, showcasing how algebra has always been a powerful tool for solving real-world problems.
In this lesson, we’ll explore the basics of algebra, including:
- Simplifying algebraic expressions
- Expanding brackets
- Solving equations where the unknown appears on both sides
Here are some key ideas to get you started:
- Algebra uses letters, often x, to represent unknown numbers.
- Algebraic expressions can be manipulated just like numerical expressions.
- x + 3 means add 3 to the unknown number.
- 3x means 3 times the unknown number.
- x² means square the unknown number.
By the end of this lesson, you’ll begin to see how algebra helps us solve problems and uncover the unknown. Let’s dive in and discover the power of algebra together!
Simplifying Algebraic Expressions
SKILL: REASONING
Investigate the result when you substitute various values (positive or negative) for x in both of these expressions:
Expression 1: \( x + 1 \)
Expression 2: \( \frac{x^2 + 6x + 5}{x + 5} \)
What is your conclusion? Which expression would you rather use?
Simplifying Expressions
Example 1: Simplify \( a + 3ab – 4ba \)
Solution: \( a + 3ab – 4ba = a – ab \)
Note: \( ab = ba \), so \( 3ab \) and \( -4ba \) are like terms and can be simplified.
Example 2: Simplify \( 3p^3 + 2p^2 – 2p^3 + 5p^2 \)
Solution: \( 3p^3 + 2p^2 – 2p^3 + 5p^2 = 3p^3 – 2p^3 + 5p^2 + 2p^2 = p^3 + 7p^2 \)
Key Notes:
- You can only add or subtract like terms.
- \( 3ab + 2ab = 5ab \), but the terms in \( 3ab + b \) cannot be added together.
- \( 3a^2 + 2a^2 = 5a^2 \), but the terms in \( 3a^2 + 2a \) cannot be added together.
- You can check your simplifications by substituting numbers.
Practice
Simplifying Algebraic Expressions – Exercises
Simplify the following expressions as much as possible.
Set 1
- \( 9ab – 5ab \)
- \( 5xy + 2yx \)
- \( 4pq – 7qp \)
- \( 2xy + y – 3xy \)
- \( x – 3x + 2 – 4x \)
- \( 7cd – 8dc + 3cd \)
- \( 6xy – 12xy + 2xy \)
- \( 4ab + 10bc – 2ab – 5cb \)
- \( 3ba – ab + 3ab – 5ab \)
- \( 4gh – 5jk – 2gh + 7 \)
- \( 2p^2 – 5p^2 + 2p – 4p \)
- \( 2x^2y – xy^2 + 3yx^2 – 2y^2x \)