Introduction to Algebraic Expressions

TL;DR

Algebraic expressions combine numbers, variables, and operation signs to represent relationships or quantities. Variables are symbols, often letters, that stand in for unknown values. Understanding these expressions is key to solving problems where values aren't fixed.

1. The Mental Model

Think of an algebraic expression as a recipe. It tells you what ingredients (numbers and variables) to use and what actions (operations) to perform, but it doesn't actually bake the cake—it just gives the instructions.

2. The Core Material

An algebraic expression is a mathematical phrase that can contain numbers, variables (like 'x' or 'y'), and operation signs (+, -, ×, ÷). Unlike an equation, an expression doesn't have an equals sign (=) and therefore doesn't state a complete thought or show a balance. It's just a way to represent a value that might change. For example, 3x + 5 is an expression.

Variables, Constants, and Coefficients

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• A variable is a symbol, usually a letter, that represents an unknown or changing value. If you're counting toys, 't' could be the variable for the number of toys.
• A constant is a number on its own, whose value doesn't change. In 3x + 5, the 5 is a constant.
• A coefficient is a number multiplied by a variable. In 3x + 5, the 3 is the coefficient of x. It tells you how many 'x's you have.

Terms

An expression is made up of terms, which are separated by addition (+) or subtraction (-) signs. Each term can be a single number, a single variable, or a product of numbers and variables.

For example, in the expression 4y - 7 + 2xy, you have three terms:
* 4y (coefficient 4, variable y)
* -7 (constant term)
* 2xy (coefficient 2, variables x and y)

graph TD
    A["Algebraic Expression"] --> B["Made of Terms"]
    B --> C["Term 1 (e.g., '3x')"]
    B --> D["Term 2 (e.g., '+ 5')"]
    C --> C1["Coefficient (e.g., '3')"]
    C --> C2["Variable (e.g., 'x')"]
    D --> D1["Constant (e.g., '5')"]

Evaluating Expressions

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"Evaluating" an expression means finding its numerical value when you're given specific values for the variables. You substitute the numbers for the variables and then perform the operations following the order of operations (PEMDAS/BODMAS).

Example: Evaluating an Expression

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Let's say you have the expression 2a + 3b - 1.
If a = 4 and b = 2:

1. Replace a with 4: 2(4) + 3b - 1
2. Replace b with 2: 2(4) + 3(2) - 1
3. Perform multiplication: 8 + 6 - 1
4. Perform addition/subtraction from left to right: 14 - 1
5. Result: 13

3. Worked Example

Imagine you're trying to figure out the cost of ordering pizzas. Each pizza costs $12, and there's a $5 delivery fee, regardless of how many pizzas you order.

We can write an algebraic expression to represent the total cost.

Let p represent the number of pizzas you order.

• The cost of the pizzas depends on how many you buy: 12 * p or 12p.
• The delivery fee is always $5.

So, the total cost expression is 12p + 5.

Now, let's evaluate this expression if you order 3 pizzas (so p = 3):

1. Substitute p with 3: 12(3) + 5
2. Perform the multiplication: 36 + 5
3. Perform the addition: 41

So, ordering 3 pizzas would cost you $41.

4. Key Takeaways

• Algebraic expressions are math phrases with numbers, variables, and operations, but no equals sign.
• Variables are symbols that represent unknown or changing values.
• A coefficient is a number multiplying a variable; a constant is a number on its own.
• Terms are parts of an expression separated by plus or minus signs.
• To evaluate an expression, substitute given values for variables and follow the order of operations.
• Expressions are tools to describe situations where values might change.
• Understanding expressions is fundamental to solving more complex algebraic problems.

Common Mistakes to Avoid:
* Confusing an expression with an equation; remember, no '=' means it's an expression.
* Ignoring the order of operations when evaluating an expression.
* Forgetting that a number next to a variable (e.g., 3x) means multiplication.
* Not correctly identifying terms, especially when subtraction is involved (e.g., x - 5 has terms x and -5).

5. Now Try It

You're saving money, and you start with $50. You plan to save an additional $15 each week.
1. Write an algebraic expression that represents the total amount of money you'll have after w weeks.
2. Evaluate your expression to find out how much money you'll have after 7 weeks.

1. Your expression should be a single line of algebraic symbols and numbers.
2. Your evaluated result should be a single number representing dollars.