Algebraic Foundations

TL;DR

Algebra is like a puzzle where you use letters to represent unknown numbers. You'll learn how to follow rules to move these numbers around and solve for the unknowns. Mastering these basics makes harder math much easier to understand.

1. The Mental Model

Think of algebra as a balancing scale. Whatever you do to one side, you must do to the other to keep it balanced. Our goal is always to get the unknown ("x") by itself.

2. The Core Material

What's a Variable?

A variable is just a letter (like x, y, or a) that stands in for an unknown number. We use them because we don't know the number yet, or it could be any number in a set.

Expressions vs. Equations

• An expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, division). It doesn't have an equals sign.
  • Examples: 3x + 5, y - 7, 2ab
• An equation is like a full sentence in math; it states that two expressions are equal. It always has an equals sign =.
  • Examples: 3x + 5 = 11, y - 7 = 2, 2ab = 12
    The whole point of an equation is usually to find the value of the variable that makes the statement true.

Basic Operations and Inverse Operations

To solve an equation, we use inverse operations to "undo" what's been done to the variable.

• Addition (+) and Subtraction (-) are inverse operations.
  • If you have x + 3 = 8, to get x alone, you subtract 3 from both sides: x + 3 - 3 = 8 - 3, so x = 5.
  • If you have x - 4 = 10, to get x alone, you add 4 to both sides: x - 4 + 4 = 10 + 4, so x = 14.
• Multiplication (× or juxtaposition) and Division (÷ or /) are inverse operations.
  • If you have 2x = 10 (which means 2 * x), to get x alone, you divide both sides by 2: 2x / 2 = 10 / 2, so x = 5.
  • If you have x / 3 = 4, to get x alone, you multiply both sides by 3: (x / 3) * 3 = 4 * 3, so x = 12.

The Order of Operations (PEMDAS/BODMAS)

When you have multiple operations, you need a specific order to solve or simplify:
1. Parentheses (or Brackets)
2. Exponents (or Orders/Powers)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

This order is critical for evaluating expressions and simplifying parts of equations correctly.

3. Worked Example

Let's solve the equation: 3x + 7 = 19

1. Isolate the term with 'x'. The +7 is with the 3x. To move it to the other side, do the inverse operation: subtract 7 from both sides.
   3x + 7 - 7 = 19 - 7
3x = 12

2. Isolate 'x'. The 3 is multiplying x. To get x alone, do the inverse operation: divide both sides by 3.
   3x / 3 = 12 / 3
x = 4

3. Check your answer. Substitute x = 4 back into the original equation:
   3(4) + 7 = 19
12 + 7 = 19
19 = 19
   It works! So x = 4 is the correct solution.

4. Key Takeaways

• Variables are letters that represent unknown numbers in math problems.
• Equations use an equals sign to show that two expressions have the same value.
• To solve an equation, you need to isolate the variable by using inverse operations.
• Whatever you do to one side of an equation, you must do to the other side.
• Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.

Common Mistakes to Avoid

• Dividing or multiplying only one part of a side instead of the entire side.
• Forgetting to apply an operation to both sides of the equation.
• Confusing the order of operations, especially multiplication/division and addition/subtraction.
• Mistaking an expression for an equation (no = sign means you can't "solve" it for a variable).

5. Now Try It

Solve the following equation for y:
5y - 12 = 18

What to do:
1. First, deal with the addition/subtraction part to get the 5y term by itself.
2. Then, use multiplication/division to get y by itself.
3. Once you have a value for y, plug it back into the original equation to check your work.

What success looks like: You should arrive at a single numerical value for y that makes the equation true when you check it.