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Introduction to Algebraic Thinking

1. What is Algebra?

Algebra is like a puzzle—instead of using only numbers, we use letters and symbols to represent unknown values. It helps us solve problems in an easy and logical way.

Real-Life Example:

Imagine you go to a shop and buy a bottle of juice and a packet of chips. The total bill is $10, but you don’t know the price of each item. If the juice costs x and the chips cost y, we can write:

x + y = 10

This equation expresses the problem using numbers and symbols.

2. Understanding Variables and Constants

Variable: A letter (like x, y, or z) representing an unknown number.

Example: In the equation x + 5 = 10, x is the variable.

Constant: A number that does not change.

Example: In x + 5 = 10, the number 5 is a constant.

Imagine a fruit basket: if the number of apples is a (variable) and there are always 4 bananas (constant), then a can change but 4 remains the same.

3. Writing and Simplifying Basic Expressions

An expression is a mathematical phrase that may contain numbers, variables, and operations (+, -, ×, ÷).

Example: 3x + 7 is an algebraic expression.

Simplifying:

  • Combine like terms: 2x + 3x = 5x
  • Arithmetic: x + 2 + 5 = x + 7

Fun Example: If you have x slices of pizza and get 3 more, the expression is x + 3. For example, if x = 5, then 5 + 3 = 8 🍕

4. Understanding the Order of Operations (PEMDAS)

PEMDAS defines the order to solve operations:

  • P → Parentheses first
  • E → Exponents
  • MD → Multiplication & Division (left-to-right)
  • AS → Addition & Subtraction (left-to-right)

Without PEMDAS: 5 + 3 × 2 → (5 + 3) = 8, then 8 × 2 = 16 ❌

With PEMDAS: First, 3 × 2 = 6, then 5 + 6 = 11 ✅

Memory Trick: "Please Excuse My Dear Aunt Sally"

5. Evaluating Expressions by Substitution

Substitution means replacing a variable with a number.

Example 1: If x = 4, then 2x + 5 becomes 2(4) + 5 = 13.

Example 2: If y = 2, then 3y - 1 becomes 3(2) - 1 = 5.

Challenge: If a = 5, solve 4a + 7.

6. Real-Life Applications of Algebra

  • ✔ Shopping: Calculating costs.
  • ✔ Cooking: Adjusting recipes.
  • ✔ Sports: Determining speeds.
  • ✔ Finance: Managing budgets.

Practice Questions (Basic Level) 🎯

  1. If x = 3, solve 2x + 4.
  2. Simplify: 5a + 2a - 3.
  3. Apply PEMDAS: (6 + 4) ÷ 2 × 3.
  4. Write an expression: You have x chocolates, and you eat 3.
  5. If y = 10, find y - 7.
  6. Solve for x: x + 5 = 12.
  7. Simplify: 4b + 3b - 2 + 6.
  8. Find 2x - 3 when x = 7.
  9. Write an algebraic expression for “5 more than twice a number n.”
  10. Solve: (8 - 3) × 2 + 4.

Want a Bigger Challenge? 🤔🔥

  1. If x = 4 and y = 3, evaluate 2x + 3y - 5.
  2. A train travels at s km/h. Write an expression for the distance traveled in t hours.
  3. If 2x + 3 = 11, solve for x.
  4. The sum of three consecutive numbers is 18. Write and solve an equation to find them.
  5. If a = 2b - 5, find a when b = 4.

🎉 Great job! Ready for more? Click below to move on to the next topic: Basic Linear Equations! 🚀

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