Algebraic Expressions and Equations

TL;DR

Algebra helps you solve problems where you don't know some numbers by using letters to stand in for them. You'll learn the difference between expressions (which are phrases) and equations (which are full sentences with an equals sign) and how to work with both. Master collecting like terms, expanding brackets, and solving basic equations to find those unknown values.

1. The Mental Model

Think of algebra like a puzzle where some pieces are missing, and you use clues to figure out what they are. You're trying to describe relationships or solve for hidden numbers using a special shorthand.

2. The Core Material

Algebra is all about using letters (variables) to represent unknown numbers. This lets you write down mathematical relationships and solve for the values you don't know.

Expressions vs. Equations

It's important to know the difference:

• Algebraic Expression: This is a mathematical "phrase" made up of numbers, variables, and mathematical operations (+, -, ×, ÷). It doesn't have an equals sign. You can simplify expressions, but you can't "solve" them for a specific value of the variable unless you're given more information.
  • Examples: 3x + 5, 2y - 7z, a/4 + b
• Algebraic Equation: This is a mathematical "sentence" that states two expressions are equal to each other. It always has an equals sign. The goal with an equation is often to find the value(s) of the variable(s) that make the equation true.
  • Examples: 3x + 5 = 11, 2y - 7z = 10, a/4 + b = 9

Here's a simple way to visualize the relationship:

graph TD
    A["Algebraic Concept"] --> B["Expression"];
    B --> C["Operators (+, -, *, /)"];
    B --> D["Variables (x, y, a)"];
    B --> E["Numbers (Coefficients, Constants)"];
    A --> F["Equation"];
    F --> B;
    F --> G["Equals Sign (=)"];

Collecting Like Terms

When you're simplifying an expression, you can only combine terms that have the exact same variable part. These are called "like terms."

• Like terms: 2x and 5x, 3y^2 and 8y^2, 4ab and -2ab.
• Unlike terms: 2x and 5y, 3y^2 and 8y, 4ab and 2a.

Rule: Add or subtract the numerical coefficients of like terms, but keep the variable part exactly the same.

Example:
Simplify 3x + 5 + 2x - 2
1. Identify like terms: (3x and 2x), (5 and -2).
2. Group them: (3x + 2x) + (5 - 2)
3. Combine: 5x + 3

Expanding Brackets

This involves multiplying a term outside the bracket by every term inside the bracket. Remember to pay attention to signs!

Example:
Expand 3(x + 4)
1. Multiply 3 by x: 3 * x = 3x
2. Multiply 3 by 4: 3 * 4 = 12
3. Combine: 3x + 12

Example with negatives:
Expand -2(y - 5)
1. Multiply -2 by y: -2 * y = -2y
2. Multiply -2 by -5: -2 * -5 = +10 (negative times negative is positive)
3. Combine: -2y + 10

Solving Basic Linear Equations

The goal is to isolate the variable (get it by itself) on one side of the equals sign. Whatever you do to one side of the equation, you must do to the other side to keep it balanced.

Core principles:
* Addition/Subtraction: To undo addition, subtract. To undo subtraction, add.
* Multiplication/Division: To undo multiplication, divide. To undo division, multiply.

Example: Solve x + 7 = 12
1. To isolate x, you need to get rid of +7.
2. Subtract 7 from both sides:
x + 7 - 7 = 12 - 7
x = 5

Example: Solve 3x = 18
1. To isolate x, you need to get rid of the 3 that's multiplying x.
2. Divide both sides by 3:
3x / 3 = 18 / 3
x = 6

Example: Solve y/4 = 5
1. To isolate y, you need to get rid of the 4 that's dividing y.
2. Multiply both sides by 4:
y/4 * 4 = 5 * 4
y = 20

3. Worked Example

Let's solve a slightly more complex equation that combines a few of these skills.

Problem: Solve 4(x + 2) - 3 = 17

Step 1: Expand the bracket.
4 * x + 4 * 2 - 3 = 17
4x + 8 - 3 = 17

Step 2: Collect like terms on the left side.
4x + (8 - 3) = 17
4x + 5 = 17

Step 3: Isolate the term with the variable (4x).
Subtract 5 from both sides:
4x + 5 - 5 = 17 - 5
4x = 12

Step 4: Isolate the variable (x).
Divide both sides by 4:
4x / 4 = 12 / 4
x = 3

Step 5 (Optional): Check your answer.
Substitute x = 3 back into the original equation:
4(3 + 2) - 3 = 17
4(5) - 3 = 17
20 - 3 = 17
17 = 17
The solution is correct!

4. Key Takeaways

• An expression is a mathematical phrase without an equals sign, while an equation is a mathematical sentence with an equals sign.
• You can only combine like terms by adding or subtracting their coefficients.
• When expanding brackets, multiply the outside term by every term inside the bracket.
• To solve an equation, perform the opposite operation to both sides to isolate the variable.
• Always perform operations in the correct order (e.g., expand brackets before collecting terms).
• Check your answers by substituting your solution back into the original equation.
• Algebra is a fundamental building block for almost all higher-level math.

Common Mistakes to Avoid:
* Forgetting to multiply all terms inside the bracket when expanding.
* Trying to combine unlike terms (e.g., 3x + 2y cannot be simplified).
* Only doing an operation to one side of an equation instead of both.
* Getting signs wrong, especially when dealing with negative numbers.

5. Now Try It

In the next 15 minutes, try to simplify the expression and then solve the equation below.

Part A (Expression): Simplify 5y - 7 + 2y + 10 - y

Part B (Equation): Solve 3(m - 4) + 6 = 15

What to do: For Part A, group and combine all like terms. For Part B, first expand the bracket, then collect any like terms, and finally use opposite operations to isolate the variable m.

What success looks like: You should have a single, simplified expression for Part A and a concrete numerical value for m in Part B.