Foundational Algebra and Equations

TL;DR

Algebra uses symbols (like letters) to represent unknown numbers in equations, letting you solve for those unknowns. The key is to keep equations balanced while you rearrange them to isolate the variable you want to find. Mastering basic operations like adding, subtracting, multiplying, and dividing on both sides of an equation is crucial.

1. The Mental Model

Think of an equation as a perfectly balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. Our goal in algebra is often to get one specific item (the unknown variable) all by itself on one side of the scale.

2. The Core Material

Algebra is essentially a powerful way to solve puzzles where some numbers are hidden. We use letters, often x or y, to stand in for these unknown numbers. An equation is a statement that two expressions are equal, shown by an = sign.

What is a Variable?

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A variable is a symbol, usually a letter, that represents a quantity that can change or is unknown. It's like a placeholder for a number you want to find.

What is an Expression vs. an Equation?

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• An expression is a mathematical phrase that can contain numbers, variables, and operations (like 2x + 5). It doesn't have an equals sign.
• An equation sets two expressions equal to each other (like 2x + 5 = 11). This is what we solve!

Solving One-Step Equations

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The goal is to get the variable by itself. You do this by performing the opposite operation on both sides of the equation.

Addition/Subtraction:
If you have x + 3 = 7, to get x alone, you need to undo the + 3. The opposite of addition is subtraction.
x + 3 - 3 = 7 - 3
x = 4

If you have x - 5 = 10, to get x alone, you need to undo the - 5. The opposite of subtraction is addition.
x - 5 + 5 = 10 + 5
x = 15

Multiplication/Division:
If you have 2x = 12, this means 2 * x = 12. To get x alone, you need to undo the * 2. The opposite of multiplication is division.
2x / 2 = 12 / 2
x = 6

If you have x / 4 = 3, to get x alone, you need to undo the / 4. The opposite of division is multiplication.
x / 4 * 4 = 3 * 4
x = 12

Solving Two-Step Equations

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These equations require two operations to isolate the variable. The general rule is to undo addition/subtraction first, then undo multiplication/division. Think of it as reversing the order of operations (PEMDAS/BODMAS).

graph TD
    A["Start with an Equation"] --> B{"Is there Addition or Subtraction with the variable?"}
    B -- "Yes" --> C["Undo Addition/Subtraction (both sides)"]
    C --> D{"Is there Multiplication or Division with the variable?"}
    B -- "No" --> D
    D -- "Yes" --> E["Undo Multiplication/Division (both sides)"]
    D -- "No" --> F["Variable is Isolated!"]
    E --> F
    F --> G["Check Your Answer"]

Example: 3x + 5 = 14
1. Undo addition: Subtract 5 from both sides.
3x + 5 - 5 = 14 - 5
3x = 9
2. Undo multiplication: Divide both sides by 3.
3x / 3 = 9 / 3
x = 3

3. Worked Example

Let's solve the equation 5x - 7 = 18.

1. Identify the variable: It's x.
2. Look for addition/subtraction near the variable: We have - 7.
3. Undo the subtraction: Add 7 to both sides of the equation.
   5x - 7 + 7 = 18 + 7
5x = 25
4. Look for multiplication/division near the variable: We have 5x (which means 5 * x).
5. Undo the multiplication: Divide both sides by 5.
   5x / 5 = 25 / 5
x = 5
6. Check your answer: Substitute x = 5 back into the original equation.
   5 * (5) - 7 = 18
25 - 7 = 18
18 = 18
   Since both sides are equal, your answer is correct!

4. Key Takeaways

• An equation is like a balanced scale; keep it balanced by doing the same thing to both sides.
• Variables are symbols (usually letters) that represent unknown numbers.
• Your main goal is to isolate the variable, getting it by itself on one side of the equals sign.
• To undo an operation, perform its opposite: add/subtract, multiply/divide.
• When solving two-step equations, undo addition/subtraction first, then multiplication/division.
• Always check your answer by plugging the solution back into the original equation.

Common Mistakes to Avoid:
- Only performing an operation on one side of the equation.
- Mixing up addition/subtraction with multiplication/division (e.g., trying to divide before subtracting).
- Forgetting that 2x means 2 * x, not 2 + x.
- Making sign errors, especially with negative numbers.

5. Now Try It

Solve the equation 4y + 11 = 3. Show your steps clearly, just like in the worked example, and then check your answer. You should be able to get the variable y by itself and verify your solution.