Algebra

Comprehensive guide to algebra covering variables, equations, functions, graphing, and algebraic problem-solving techniques.

algebra variables equations functions graphing polynomials factoring

Prerequisites

Required:

BasicMath - All arithmetic operations, fractions, order of operations (BODMAS)

Helpful:

Comfort with negative numbers and basic problem-solving

Overview

Algebra is the branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. It’s the bridge between basic arithmetic and advanced mathematics, teaching us to work with unknown values and solve complex problems systematically.

This guide progresses from basic variable concepts through equation solving, graphing, and advanced algebraic techniques. Master these fundamentals to unlock higher mathematics and practical problem-solving skills.

Variables and Expressions

Understanding Variables

A variable is a letter or symbol that represents an unknown number or a number that can change.

Examples:
• x, y, z (most common variables)
• a, b, c (often used for constants)
• n (often used for counting)
• t (often used for time)

Why Use Variables?

Instead of: "A number plus 5 equals 12"
We write: x + 5 = 12

Variables let us:
• Represent unknown quantities
• Write general formulas
• Solve problems systematically

Algebraic Expressions

An algebraic expression combines variables, numbers, and operations.

Basic Terms

Expression: 3x + 5
• 3x is a "term" (coefficient 3, variable x)
• 5 is a "constant term"
• + is the "operation"

Expression: 2x² - 4x + 7
• 2x² (coefficient 2, variable x squared)
• -4x (coefficient -4, variable x) 
• 7 (constant term)

Like Terms

Terms that have the same variable raised to the same power.

Like terms (can be combined):
• 3x and 7x → 10x
• 5y² and -2y² → 3y²
• 8 and -3 → 5

Unlike terms (cannot be combined):
• 3x and 5y (different variables)
• 2x and 4x² (different powers)

Simplifying Expressions

Combining Like Terms

Example 1: 3x + 5x - 2x
= (3 + 5 - 2)x
= 6x

Example 2: 4x² + 2x - 3x² + 7x - 1
= 4x² - 3x² + 2x + 7x - 1
= x² + 9x - 1

Distributive Property

a(b + c) = ab + ac

Examples:
3(x + 4) = 3x + 12
-2(3y - 5) = -6y + 10
x(x + 3) = x² + 3x

Complex example:
2(3x + 1) - 4(x - 2)
= 6x + 2 - 4x + 8
= 2x + 10

Evaluating Expressions

Substitute numbers for variables and calculate.

Expression: 2x + 3y - 5
If x = 4 and y = -1:
2(4) + 3(-1) - 5
= 8 - 3 - 5
= 0

Expression: x² - 4x + 3
If x = 5:
(5)² - 4(5) + 3
= 25 - 20 + 3
= 8

Equations

An equation states that two expressions are equal. Our goal is to find the value(s) of the variable that make the equation true.

Linear Equations

One-Step Equations

x + 7 = 12
Subtract 7 from both sides:
x = 5

3x = 21
Divide both sides by 3:
x = 7

x/4 = 6
Multiply both sides by 4:
x = 24

Two-Step Equations

2x + 5 = 13
Subtract 5: 2x = 8
Divide by 2: x = 4

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Multi-Step Equations

3(x - 2) + 4x = 22
Step 1: Distribute
3x - 6 + 4x = 22
Step 2: Combine like terms
7x - 6 = 22
Step 3: Add 6
7x = 28
Step 4: Divide by 7
x = 4

Check: 3(4 - 2) + 4(4) = 6 + 16 = 22 ✓

Equations with Variables on Both Sides

5x - 3 = 2x + 9
Step 1: Subtract 2x from both sides
3x - 3 = 9
Step 2: Add 3 to both sides
3x = 12
Step 3: Divide by 3
x = 4

Solving Strategy

Simplify both sides (distribute, combine like terms)
Collect variable terms on one side
Collect constants on the other side
Divide by the coefficient of the variable
Check your answer

Systems of Equations

When you have multiple equations with multiple variables.

Two-Variable Systems

Substitution Method

System:
y = 2x + 1    ... (1)
3x + y = 11   ... (2)

Step 1: Substitute (1) into (2)
3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2

Step 2: Find y using equation (1)
y = 2(2) + 1 = 5

Solution: (2, 5)

Check: 3(2) + 5 = 6 + 5 = 11 ✓

Elimination Method

System:
2x + 3y = 7   ... (1)
4x - 3y = 5   ... (2)

Step 1: Add equations (y-terms cancel)
6x = 12
x = 2

Step 2: Substitute back
2(2) + 3y = 7
4 + 3y = 7
3y = 3
y = 1

Solution: (2, 1)

Word Problems with Systems

Problem: Concert tickets cost $15 for adults and $8 for children. 
120 tickets were sold for $1350 total. How many of each type?

Let: a = adult tickets, c = child tickets

Equations:
a + c = 120        (total tickets)
15a + 8c = 1350   (total money)

From first equation: c = 120 - a
Substitute: 15a + 8(120 - a) = 1350
15a + 960 - 8a = 1350
7a = 390
a = 60

Therefore: c = 120 - 60 = 60

Answer: 60 adult tickets, 60 child tickets

Quadratic Equations

Equations where the highest power of the variable is 2.

Standard Form

ax² + bx + c = 0
where a ≠ 0

Examples:
x² - 5x + 6 = 0
2x² + 3x - 2 = 0
x² - 4 = 0

Solving Methods

Factoring

x² - 5x + 6 = 0
Factor: (x - 2)(x - 3) = 0
Solutions: x = 2 or x = 3

x² - 9 = 0
Factor: (x + 3)(x - 3) = 0
Solutions: x = -3 or x = 3

Quadratic Formula

For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / (2a)

Example: 2x² + 5x - 3 = 0
a = 2, b = 5, c = -3

x = (-5 ± √(25 - 4(2)(-3))) / (2(2))
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
x = (-5 ± 7) / 4

Solutions: x = 2/4 = 1/2 or x = -12/4 = -3

Completing the Square

x² + 6x + 5 = 0
Step 1: Move constant
x² + 6x = -5
Step 2: Complete the square
x² + 6x + 9 = -5 + 9
(x + 3)² = 4
Step 3: Take square root
x + 3 = ±2
x = -3 ± 2
Solutions: x = -1 or x = -5

Functions

A function is a rule that assigns exactly one output to each input.

Function Notation

f(x) = 2x + 3

Read as: "f of x equals 2x plus 3"
• x is the input (independent variable)
• f(x) is the output (dependent variable)

To find f(4):
f(4) = 2(4) + 3 = 11

Types of Functions

Linear Functions

f(x) = mx + b
• m is the slope
• b is the y-intercept
• Graph is a straight line

Example: f(x) = 2x - 1
Slope = 2, y-intercept = -1

Quadratic Functions

f(x) = ax² + bx + c
• Graph is a parabola
• Opens up if a > 0, down if a < 0

Example: f(x) = x² - 4x + 3
Vertex form: f(x) = (x - 2)² - 1
Vertex at (2, -1)

Exponential Functions

f(x) = abˣ
• a is initial value
• b is growth factor

Example: f(x) = 2 · 3ˣ
Starts at 2, triples each time x increases by 1

Domain and Range

Domain: All possible input values (x-values) Range: All possible output values (y-values)

f(x) = √x
Domain: x ≥ 0 (can't take square root of negative)
Range: f(x) ≥ 0 (square root is always non-negative)

f(x) = 1/x
Domain: x ≠ 0 (can't divide by zero)
Range: f(x) ≠ 0 (fraction never equals zero)

Graphing

Coordinate Plane

Points are located using (x, y) coordinates.

Quadrants:
I: (+, +)    II: (-, +)
III: (-, -)  IV: (+, -)

Important lines:
x-axis: y = 0
y-axis: x = 0
Origin: (0, 0)

Graphing Linear Equations

Slope-Intercept Form

y = mx + b
• m = slope (rise/run)
• b = y-intercept

Example: y = 2x - 3
1. Start at y-intercept: (0, -3)
2. Use slope 2 = 2/1: up 2, right 1
3. Plot points and draw line

Finding Slope

Slope = (y₂ - y₁)/(x₂ - x₁)

Between points (1, 3) and (4, 9):
m = (9 - 3)/(4 - 1) = 6/3 = 2

Graphing by Intercepts

For 2x + 3y = 6:

x-intercept (set y = 0):
2x + 3(0) = 6
x = 3
Point: (3, 0)

y-intercept (set x = 0):
2(0) + 3y = 6
y = 2
Point: (0, 2)

Draw line through both points

Graphing Quadratic Functions

f(x) = x² - 4x + 3

Method 1: Make a table
x | -1 | 0 | 1 | 2 | 3 | 4 | 5
y |  8 | 3 | 0 |-1 | 0 | 3 | 8

Method 2: Find vertex and axis of symmetry
Vertex x = -b/(2a) = -(-4)/(2·1) = 2
f(2) = 4 - 8 + 3 = -1
Vertex: (2, -1)
Axis of symmetry: x = 2

Polynomials

Polynomial Operations

Adding and Subtracting

(3x² + 2x - 1) + (x² - 4x + 3)
= 3x² + x² + 2x - 4x - 1 + 3
= 4x² - 2x + 2

(5x² - 3x + 2) - (2x² + x - 4)
= 5x² - 3x + 2 - 2x² - x + 4
= 3x² - 4x + 6

Multiplying

3x(x² - 2x + 1) = 3x³ - 6x² + 3x

(x + 2)(x + 3)
= x² + 3x + 2x + 6
= x² + 5x + 6

(2x - 1)(x² + 3x - 2)
= 2x(x² + 3x - 2) - 1(x² + 3x - 2)
= 2x³ + 6x² - 4x - x² - 3x + 2
= 2x³ + 5x² - 7x + 2

Factoring

Greatest Common Factor

6x³ + 9x² - 3x
GCF = 3x
= 3x(2x² + 3x - 1)

Factoring Quadratics

x² + 7x + 12
Find two numbers that multiply to 12 and add to 7: 3 and 4
= (x + 3)(x + 4)

2x² - 5x - 3
Find factors of ac = -6 that add to -5: -6 and 1
= 2x² - 6x + x - 3
= 2x(x - 3) + 1(x - 3)
= (2x + 1)(x - 3)

Special Patterns

Difference of squares: a² - b² = (a + b)(a - b)
x² - 9 = (x + 3)(x - 3)

Perfect square trinomials:
a² + 2ab + b² = (a + b)²
x² + 6x + 9 = (x + 3)²

a² - 2ab + b² = (a - b)²
x² - 10x + 25 = (x - 5)²

Inequalities

Similar to equations, but with inequality symbols: <, >, ≤, ≥

Solving Linear Inequalities

3x + 5 > 14
Subtract 5: 3x > 9
Divide by 3: x > 3

Solution: All numbers greater than 3
Graph: Open circle at 3, arrow to the right

Important Rule

When multiplying or dividing by a negative number, flip the inequality sign.

-2x > 6
Divide by -2 (flip sign): x < -3

Compound Inequalities

-3 < 2x + 1 ≤ 7
Subtract 1: -4 < 2x ≤ 6
Divide by 2: -2 < x ≤ 3

Solution: x is between -2 and 3 (including 3)

Absolute Value Inequalities

|x - 2| < 5
Means: -5 < x - 2 < 5
Add 2: -3 < x < 7

|x + 1| ≥ 4
Means: x + 1 ≤ -4 or x + 1 ≥ 4
Solutions: x ≤ -5 or x ≥ 3

Word Problems and Applications

Translation Guide

Phrase → Mathematical Expression
"5 more than a number" → x + 5
"3 less than a number" → x - 3
"twice a number" → 2x
"half of a number" → x/2
"5 more than twice a number" → 2x + 5
"the sum of two consecutive integers" → x + (x + 1)

Problem-Solving Strategy

Read the problem carefully
Identify what you’re looking for
Define variables
Write an equation
Solve the equation
Check if your answer makes sense
Answer the question in words

Example Problems

Age Problem

Problem: Sarah is 3 times as old as her brother. In 5 years, 
she will be twice as old as he will be. How old are they now?

Let x = brother's current age
Then 3x = Sarah's current age

In 5 years:
Brother's age: x + 5
Sarah's age: 3x + 5

Equation: 3x + 5 = 2(x + 5)
3x + 5 = 2x + 10
x = 5

Answer: Brother is 5, Sarah is 15

Distance-Rate-Time Problem

Problem: Two cars leave from the same point traveling in opposite 
directions. One travels at 60 mph, the other at 70 mph. 
After how many hours will they be 390 miles apart?

Let t = time in hours
Distance = rate × time

Car 1 distance: 60t
Car 2 distance: 70t
Total distance apart: 60t + 70t = 130t

Equation: 130t = 390
t = 3

Answer: After 3 hours

Work Problem

Problem: John can paint a fence in 4 hours. Mike can paint 
the same fence in 6 hours. How long will it take if they work together?

John's rate: 1/4 fence per hour
Mike's rate: 1/6 fence per hour
Combined rate: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 fence per hour

Time = Work ÷ Rate = 1 ÷ (5/12) = 12/5 = 2.4 hours

Answer: 2 hours 24 minutes

Advanced Topics

Rational Expressions

Simplifying

(x² - 4)/(x + 2) = (x + 2)(x - 2)/(x + 2) = x - 2
(provided x ≠ -2)

Operations

Addition: a/b + c/d = (ad + bc)/(bd)
2/x + 3/(x-1) = (2(x-1) + 3x)/(x(x-1)) = (5x-2)/(x(x-1))

Multiplication: (a/b) × (c/d) = ac/bd
(x+1)/x × x/(x+3) = (x+1)/(x+3)

Radical Expressions

Simplifying Radicals

√72 = √(36 × 2) = 6√2
∛24 = ∛(8 × 3) = 2∛3
√(x⁸) = x⁴ (assuming x ≥ 0)

Operations with Radicals

√12 + √27 = 2√3 + 3√3 = 5√3
√6 × √10 = √60 = 2√15
(√5 + 2)(√5 - 3) = 5 - 3√5 + 2√5 - 6 = -1 - √5

Exponential and Logarithmic Equations

Exponential Equations

2^x = 8
2^x = 2³
x = 3

3^(2x-1) = 27
3^(2x-1) = 3³
2x - 1 = 3
x = 2

Practice Problems

Linear Equations

1. Solve: 3(x - 2) + 4 = 2x + 5
2. Solve: 2x + 3 = 5x - 12
3. The sum of three consecutive integers is 78. Find the integers.

Systems of Equations

1. Solve: x + y = 5, 2x - y = 1
2. Solve: 3x + 2y = 12, x - y = 1
3. Two numbers have a sum of 20 and a difference of 4. Find them.

Quadratics

1. Solve by factoring: x² - 7x + 12 = 0
2. Solve using the quadratic formula: 2x² + 3x - 2 = 0
3. Find the vertex of f(x) = x² - 6x + 8

Functions

1. If f(x) = 3x - 5, find f(-2)
2. Find the domain of f(x) = √(x - 3)
3. Graph y = 2x + 1

Word Problems

1. The length of a rectangle is 3 more than twice its width. 
   If the perimeter is 36, find the dimensions.

2. A boat travels 20 miles downstream in the same time 
   it takes to travel 12 miles upstream. If the current 
   is 2 mph, find the boat's speed in still water.

Answer Key

Linear Equations

1. 3(x - 2) + 4 = 2x + 5 → x = 7
2. 2x + 3 = 5x - 12 → x = 5
3. Three consecutive integers: 25, 26, 27

Systems of Equations

1. x + y = 5, 2x - y = 1 → (2, 3)
2. 3x + 2y = 12, x - y = 1 → (2, 3)
3. Two numbers: 12 and 8

Quadratics

1. x² - 7x + 12 = 0 → x = 3 or x = 4
2. 2x² + 3x - 2 = 0 → x = 1/2 or x = -2
3. Vertex of f(x) = x² - 6x + 8 is (3, -1)

Functions

1. f(-2) = 3(-2) - 5 = -11
2. Domain: x ≥ 3
3. Graph: Line with slope 2, y-intercept 1

Word Problems

1. Rectangle: width = 5, length = 13
2. Boat speed: 8 mph in still water

Quick Reference

Equation Solving Steps

Simplify both sides
Collect variables on one side
Collect constants on the other side
Divide by coefficient
Check your answer

Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

Function Notation

f(x): function notation
Domain: all possible x-values
Range: all possible y-values

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁)

Common Factoring Patterns

a² - b² = (a + b)(a - b)
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²