Algebra Foundations Reference Sheet
Algebra Toolkit • Part 1 Visual Reference
Order of Operations: GEMDAS
Follow Top to Bottom
Use this roadmap to simplify math expressions step-by-step without getting lost.
The 4 Priority Levels
G Level 1
Grouping
Innermost to outermost symbols first.
( ) [ ] | | top/bot
E Level 2
Exponents
Powers and square root radicals.
\(3^2 = 9\) • \(\sqrt{25} = 5\)
M D
Level 3
Multiply & Divide
Equal rank! Whichever appears first.
Left Right
A S
Level 4
Add & Subtract
Equal rank! Whichever appears first.
Left Right
Step-by-Step Master Example
Expression: \( 15 - 3 \cdot (2 + 2)^2 \div 8 \)
Step 1: G Inside parentheses: \(2 + 2 = 4\)
\( 15 - 3 \cdot \) \((4)^2\) \( \div 8 \)
Step 2: E Calculate exponent: \(4^2 = 16\)
\( 15 - 3 \cdot \) \(16\) \( \div 8 \)
Step 3: M Multiply & Divide left-to-right: \(3 \cdot 16 = 48\)
\( 15 - \) \(48\) \( \div 8 \)
Step 4: D Finish division next: \(48 \div 8 = 6\)
\( 15 - \) \(6\)
Step 5: S Subtract: \(15 - 6 = 9\)
Final Answer = 9
TRAP: Left-to-Right Tie-Breaker
Multiplication does NOT always beat division! Work whichever comes first from left to right.
\(12 \div 3 \cdot 2 = 4 \cdot 2 = 8\) \(12 \div 6 = 2\)
PRO-TIP: Negative Exponent Trap
Look closely at parentheses around negative numbers before squaring!
\((-4)^2 = (-4)(-4) = \mathbf{+16}\)
|
\(-4^2 = -(4 \cdot 4) = \mathbf{-16}\)
Algebra Toolkit • Part 2 Visual Reference
Combining Like Terms
Match Variables & Powers
Terms can only be combined if they have the exact same variable raised to the exact same power.
Anatomy of an Algebraic Term: \(-7x^2\)
-
The Sign
Always travels with the number right behind it!
7
Coefficient
The multiplier number in front of the letter.
x
Variable
The letter representing the unknown value.
\(^2\)
Exponent
The power. Must match identically to combine!
YES! LIKE TERMS (CAN COMBINE)
\(5x\) and \(-2x\) Both have \(x^1\)
\(4y^2\) and \(9y^2\) Both have \(y^2\)
\(12\) and \(-5\) Plain constants
NO! UNLIKE TERMS (CANNOT COMBINE)
\(3x\) and \(3y\) Different variables!
\(5x\) and \(5x^2\) Different exponents!
\(8x\) and \(8\) One variable, one plain
The "Shape & Color" Strategy Rule: Include the sign in front!
Original Expression:
\(5x\) + \(3\) - \(2x\) + \(4y\) + \(6\)
1. Boxes: \(x\)-terms
\(5x - 2x = \mathbf{3x}\)
Combine \(5 - 2\)
2. Dash: \(y\)-terms
\(\mathbf{+4y}\)
No other \(y\) term, keep as is!
3. Circles: Numbers
\(+3 + 6 = \mathbf{+9}\)
Regular constants
Simplified Form: \(3x + 4y + 9\) Fully Simplified!
Integer Sign Helper (When Combining Coefficients)
SAME SIGNS: Add & Keep
Add the numbers, keep their sign.
\(-4x - 5x = \mathbf{-9x}\) • \(3x + 2x = \mathbf{5x}\)
DIFFERENT SIGNS: Subtract & Take Larger
Subtract numbers, keep sign of bigger absolute value.
\(-7x + 2x = \mathbf{-5x}\) • \(8x - 3x = \mathbf{5x}\)
