Algebraic Mastery

TL;DR

Algebraic mastery is about confidently manipulating equations to solve for unknowns, using foundational skills like expanding, factorising, and simplifying expressions. It builds on understanding how operations balance across an equals sign, allowing you to rearrange formulas effectively. By breaking down complex problems into smaller steps, you can tackle even challenging algebraic tasks.

1. The Mental Model

Think of an equation like a balanced scale: whatever you do to one side, you must do to the exact same thing to the other side to keep it balanced. Your goal is often to isolate a specific variable, moving everything else away from it while maintaining that balance.

2. The Core Material

Expanding Expressions

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Expanding means multiplying terms inside brackets by terms outside or by other terms in different brackets. It's like distributing a value.

Example:
* $2(x + 3) = 2x + 6$
* $(x + 1)(x + 2) = x(x + 2) + 1(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2$

Factorising Expressions

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Factorising is the reverse of expanding. You're looking for common factors to pull out of terms or turning an expression back into a product of brackets.

Types of Factorising:
1. Common Factor: Find a number or variable that divides into all terms.
* $3x + 6 = 3(x + 2)$
* $5xy - 10y = 5y(x - 2)$
2. Difference of Two Squares (DOTS): If you have $a^2 - b^2$, it always factorises to $(a - b)(a + b)$.
* $x^2 - 9 = (x - 3)(x + 3)$
* $4y^2 - 25 = (2y - 5)(2y + 5)$
3. Trinomials (Quadratic Expressions): For expressions like $ax^2 + bx + c$, you look for two numbers that multiply to 'ac' and add to 'b'.

```mermaid
graph TD
    A["Start: Trinomial ax² + bx + c"] --> B{"Find two numbers, p and q"}
    B -- "p * q = ac" --> C{"Find two numbers, p and q"}
    B -- "p + q = b" --> C
    C --> D["Rewrite bx as px + qx"]
    D --> E["Factorise by grouping"]
    E --> F["Result: Two brackets multiplied"]
```

**Example:** Factorise $x^2 + 5x + 6$
*   'ac' = $1 \times 6 = 6$
*   'b' = $5$
*   Numbers that multiply to 6 and add to 5 are 2 and 3.
*   $x^2 + 2x + 3x + 6$
*   $x(x + 2) + 3(x + 2)$
*   $(x + 2)(x + 3)$

Simplifying Expressions

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Simplifying means combining like terms. Like terms have the exact same variables raised to the exact same powers.

Example:
* $3x + 5y - x + 2y = (3x - x) + (5y + 2y) = 2x + 7y$
* $4x^2 + 2x - x^2 + 7 = (4x^2 - x^2) + 2x + 7 = 3x^2 + 2x + 7$

Solving Equations

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This is about finding the value of the unknown variable that makes the equation true. Remember the 'balanced scale' rule.

Example: Solve for $x$ in $3x + 5 = 14$
1. Subtract 5 from both sides: $3x + 5 - 5 = 14 - 5 \implies 3x = 9$
2. Divide both sides by 3: $3x / 3 = 9 / 3 \implies x = 3$

Rearranging Formulas

This involves using the same techniques as solving equations, but your goal isn't to find a numerical answer, but to make a different variable the subject of the formula.

Example: Make 'a' the subject of $v = u + at$
1. Subtract 'u' from both sides: $v - u = at$
2. Divide by 't' from both sides: $(v - u) / t = a$
So, $a = (v - u) / t$

3. Worked Example

Let's solve for $x$ in the equation: $3(x + 2) - 2(x - 1) = 10$

1. Expand the brackets:
   $3x + 6 - (2x - 2) = 10$
   Remember to distribute the negative sign to all terms inside the second bracket!
   $3x + 6 - 2x + 2 = 10$

2. Combine like terms on the left side:
   $(3x - 2x) + (6 + 2) = 10$
   $x + 8 = 10$

3. Isolate x (subtract 8 from both sides):
   $x + 8 - 8 = 10 - 8$
   $x = 2$

4. Key Takeaways

• Always keep equations balanced by performing the same operation on both sides.
• Expanding involves multiplying terms to remove brackets.
• Factorising is the reverse of expanding, turning expressions into products.
• Simplify by combining only "like terms" (same variable, same power).
• When rearranging formulas, treat the variables like numbers you're moving around.
• Pay close attention to negative signs, especially when expanding or dealing with grouped terms.
• Practice identifying the type of factorising needed for different expressions.

5. Now Try It

Solve the following equation for $y$: $5(y - 3) + 2y = 4(y + 1)$

Success looks like: Arriving at $y = 19/3$ by first expanding all brackets, then collecting like terms on each side of the equation, and finally isolating $y$.