Grade 9 Exam > Grade 9 Notes > Integrated Math 1 > Cheatsheet: Algebra Foundations
Cheatsheet: Algebra Foundations
1. Real Numbers and Properties
1.1 Number Sets
| Number Set | Definition and Examples |
|---|---|
| Natural Numbers (ℕ) | Counting numbers: {1, 2, 3, 4, ...} |
| Whole Numbers (W) | Natural numbers and zero: {0, 1, 2, 3, ...} |
| Integers (ℤ) | Whole numbers and their negatives: {..., -2, -1, 0, 1, 2, ...} |
| Rational Numbers (ℚ) | Numbers that can be written as a/b where a, b are integers and b ≠ 0; includes terminating and repeating decimals |
| Irrational Numbers | Numbers that cannot be written as fractions; non-terminating, non-repeating decimals (√2, π, √3) |
| Real Numbers (ℝ) | All rational and irrational numbers combined |
1.2 Properties of Real Numbers
| Property | Definition |
|---|---|
| Commutative Property (Addition) | a + b = b + a |
| Commutative Property (Multiplication) | a × b = b × a |
| Associative Property (Addition) | (a + b) + c = a + (b + c) |
| Associative Property (Multiplication) | (a × b) × c = a × (b × c) |
| Distributive Property | a(b + c) = ab + ac |
| Identity Property (Addition) | a + 0 = a |
| Identity Property (Multiplication) | a × 1 = a |
| Inverse Property (Addition) | a + (-a) = 0 |
| Inverse Property (Multiplication) | a × (1/a) = 1, where a ≠ 0 |
| Zero Property of Multiplication | a × 0 = 0 |
2. Order of Operations
2.1 PEMDAS/GEMDAS
- P: Parentheses (and other grouping symbols like brackets, braces)
- E: Exponents (powers and roots)
- M/D: Multiplication and Division (left to right)
- A/S: Addition and Subtraction (left to right)
2.2 Key Rules
- Operations at the same level are performed left to right
- Multiplication does not always come before division
- Addition does not always come before subtraction
- Evaluate innermost grouping symbols first
3. Variables and Expressions
3.1 Terminology
| Term | Definition |
|---|---|
| Variable | A letter or symbol that represents an unknown value |
| Constant | A fixed numerical value |
| Coefficient | The numerical factor of a term containing a variable (in 5x, 5 is the coefficient) |
| Term | A single number, variable, or product of numbers and variables (3, x, 5xy) |
| Expression | A mathematical phrase combining numbers, variables, and operations (3x + 5) |
| Equation | A mathematical statement showing two expressions are equal (3x + 5 = 11) |
3.2 Types of Expressions
| Type | Definition |
|---|---|
| Monomial | Expression with one term (5x²) |
| Binomial | Expression with two terms (3x + 7) |
| Trinomial | Expression with three terms (x² + 5x + 6) |
| Polynomial | Expression with one or more terms (general term) |
3.3 Like Terms
- Terms with identical variable parts (same variables raised to same powers)
- Examples: 3x and 7x are like terms; 3x² and 7x are not like terms
- Combine like terms by adding or subtracting coefficients
- Example: 5x + 3x = 8x
4. Evaluating Expressions
4.1 Substitution Process
- Replace each variable with its given value
- Use parentheses around substituted values
- Follow order of operations to simplify
- Example: Evaluate 3x + 5 when x = 4 → 3(4) + 5 = 12 + 5 = 17
5. Simplifying Expressions
5.1 Steps for Simplification
- Remove parentheses using distributive property
- Combine like terms
- Write in standard form (descending order of exponents)
5.2 Distributive Property Applications
- a(b + c) = ab + ac
- a(b - c) = ab - ac
- -(a + b) = -a - b
- -(a - b) = -a + b
6. Exponent Rules
6.1 Basic Exponent Laws
| Rule | Formula |
|---|---|
| Product of Powers | a^m × a^n = a^(m+n) |
| Quotient of Powers | a^m ÷ a^n = a^(m-n), where a ≠ 0 |
| Power of a Power | (a^m)^n = a^(mn) |
| Power of a Product | (ab)^n = a^n × b^n |
| Power of a Quotient | (a/b)^n = a^n/b^n, where b ≠ 0 |
| Zero Exponent | a^0 = 1, where a ≠ 0 |
| Negative Exponent | a^(-n) = 1/a^n, where a ≠ 0 |
| Identity | a^1 = a |
7. Linear Equations in One Variable
7.1 Solving One-Step Equations
| Type | Method |
|---|---|
| Addition Equation (x + a = b) | Subtract a from both sides |
| Subtraction Equation (x - a = b) | Add a to both sides |
| Multiplication Equation (ax = b) | Divide both sides by a |
| Division Equation (x/a = b) | Multiply both sides by a |
7.2 Solving Multi-Step Equations
- Step 1: Simplify each side (distribute, combine like terms)
- Step 2: Move variable terms to one side using addition/subtraction
- Step 3: Move constant terms to the other side
- Step 4: Divide or multiply to isolate the variable
- Step 5: Check solution by substituting back into original equation
7.3 Equations with Variables on Both Sides
- Collect all variable terms on one side
- Collect all constants on the other side
- Solve for the variable
- Example: 3x + 5 = 2x + 9 → x + 5 = 9 → x = 4
7.4 Special Cases
| Type | Characteristic |
|---|---|
| Identity (Infinitely Many Solutions) | Results in a true statement (5 = 5); all real numbers are solutions |
| Contradiction (No Solution) | Results in a false statement (3 = 7); no values satisfy the equation |
8. Literal Equations and Formulas
8.1 Definition
- Equations with two or more variables
- Solving for one variable in terms of the others
8.2 Common Formulas
| Formula | Description |
|---|---|
| P = 2l + 2w | Perimeter of rectangle (l = length, w = width) |
| A = lw | Area of rectangle |
| A = ½bh | Area of triangle (b = base, h = height) |
| d = rt | Distance formula (r = rate, t = time) |
| C = 2πr | Circumference of circle (r = radius) |
| A = πr² | Area of circle |
8.3 Solving Literal Equations
- Use same steps as solving regular equations
- Isolate the specified variable
- Example: Solve A = lw for w → w = A/l
9. Ratios, Rates, and Proportions
9.1 Ratios
- Comparison of two quantities: a:b or a/b
- Can be simplified like fractions
- Example: 6:8 simplifies to 3:4
9.2 Rates
- Ratio comparing quantities with different units
- Unit rate: rate with denominator of 1
- Example: 120 miles in 2 hours = 60 miles per hour
9.3 Proportions
- Equation stating two ratios are equal: a/b = c/d
- Cross-multiplication: ad = bc
- Used to solve for unknown values
- Example: 3/4 = x/12 → 4x = 36 → x = 9
10. Percent
10.1 Percent Basics
- Percent means "per hundred" (represented by %)
- Convert percent to decimal: divide by 100 (25% = 0.25)
- Convert decimal to percent: multiply by 100 (0.6 = 60%)
- Convert fraction to percent: divide, then multiply by 100
10.2 Percent Equation
- Part = Percent × Whole
- Percent = Part/Whole
- Whole = Part/Percent
10.3 Percent Applications
| Application | Formula |
|---|---|
| Percent Increase | ((New - Original)/Original) × 100% |
| Percent Decrease | ((Original - New)/Original) × 100% |
| Markup/Discount | New Price = Original ± (Percent × Original) |
| Sales Tax | Total = Price + (Tax Rate × Price) |
11. Absolute Value
11.1 Definition
- Distance from zero on number line (always non-negative)
- |a| = a if a ≥ 0; |a| = -a if a <>
- Example: |5| = 5, |-5| = 5
11.2 Properties
- |a| ≥ 0 for all real numbers a
- |a| = |-a|
- |ab| = |a| × |b|
- |a/b| = |a|/|b|, where b ≠ 0
12. Inequalities
12.1 Inequality Symbols
| Symbol | Meaning |
|---|---|
| > | Greater than |
| Less than | |
| ≥ | Greater than or equal to |
| ≤ | Less than or equal to |
| ≠ | Not equal to |
12.2 Solving Linear Inequalities
- Use same steps as solving equations
- Key rule: When multiplying or dividing by a negative number, reverse the inequality sign
- Example: -2x > 6 → x < -3="" (sign="">
- Solution represented on number line or in interval notation
12.3 Graphing on Number Line
- Open circle (○): < or=""> (value not included)
- Closed circle (●): ≤ or ≥ (value included)
- Shade to the right for > or ≥
- Shade to the left for < or="">
12.4 Interval Notation
| Inequality | Interval Notation |
|---|---|
| x > a | (a, ∞) |
| x ≥ a | [a, ∞) |
| x <> | (-∞, b) |
| x ≤ b | (-∞, b] |
| a < x=""><> | (a, b) |
| a ≤ x ≤ b | [a, b] |
13. Coordinate Plane
13.1 Components
- x-axis: horizontal number line
- y-axis: vertical number line
- Origin: point (0, 0) where axes intersect
- Ordered pair: (x, y) where x is horizontal position, y is vertical position
13.2 Quadrants
| Quadrant | Signs |
|---|---|
| I | (+, +) |
| II | (-, +) |
| III | (-, -) |
| IV | (+, -) |
14. Relations and Functions
14.1 Definitions
| Term | Definition |
|---|---|
| Relation | Set of ordered pairs (x, y) |
| Domain | Set of all x-values (input values) |
| Range | Set of all y-values (output values) |
| Function | Relation where each input has exactly one output |
14.2 Vertical Line Test
- Graph represents a function if no vertical line intersects it more than once
- Used to determine if a graph is a function
14.3 Function Notation
- f(x) read as "f of x" or "the value of f at x"
- f(x) represents the output when x is the input
- Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11
15. Linear Functions
15.1 Slope
- Measure of steepness of a line
- m = (y₂ - y₁)/(x₂ - x₁) = rise/run
- Positive slope: line rises left to right
- Negative slope: line falls left to right
- Zero slope: horizontal line
- Undefined slope: vertical line
15.2 Forms of Linear Equations
| Form | Equation |
|---|---|
| Slope-Intercept Form | y = mx + b (m = slope, b = y-intercept) |
| Point-Slope Form | y - y₁ = m(x - x₁) (m = slope, (x₁, y₁) = point on line) |
| Standard Form | Ax + By = C (A, B, C are integers, A ≥ 0) |
15.3 Intercepts
| Intercept | Definition |
|---|---|
| x-intercept | Point where line crosses x-axis (y = 0) |
| y-intercept | Point where line crosses y-axis (x = 0) |
15.4 Special Lines
| Type | Equation |
|---|---|
| Horizontal Line | y = k (slope = 0) |
| Vertical Line | x = h (undefined slope) |
| Parallel Lines | Same slope, different y-intercepts (m₁ = m₂) |
| Perpendicular Lines | Slopes are negative reciprocals (m₁ × m₂ = -1) |
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