Algebra Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in algebra.

Fundamentals - Theory & Concepts

Review of basic algebraic concepts including order of operations, real number properties, absolute value, and inequalities.

Properties of Real Numbers on the Number Line

Select Property

Value of a3

Value of b-2

Commutative Property

a + b = b + a

3 + (-2) = -2 + (3)

1 = 1

Visualize

a + b

and

b + a

reaching the same sum.

Fundamentals - Theory & Concepts - Absolute Value

Review of basic algebraic concepts including order of operations, real number properties, absolute value, and inequalities.

Absolute Value Equations & Distance Explorer

Problem Mode

Coefficient a2

Constant b-4

Distance Limit c6

Step-by-Step Cases

Case 1 (+)

ax + b = c \implies 2x - 4 = 6

x = \frac{6 - (-4)}{2} = 5.0

Case 2 (-)

ax + b = -c \implies 2x - 4 = -6

x = \frac{-6 - (-4)}{2} = -1.0

Mathematical State

|2x - 4| = 6

Computed Solution Set

x \in \{-1.00, 5.00\}

Exponents and Radicals - Theory & Concepts - Exponent Rules

Detailed guide on the laws of exponents, simplifying radicals, rationalizing denominators, and solving radical equations.

Laws of Exponents Sandbox

Select Exponent Law

a^m \cdot a^n = a^{m+n}

Base (a)2

Exponent (m)2

Exponent (n)3

Active Proof Expansion

a^m \cdot a^n = a^{m+n}

2^2 \cdot 2^3 = 2^{2+3} = 32

Visualizing multiplication (matching bases)

a^m (2^2)

(2 · 2)

a^n (2^3)

(2 · 2 · 2)

⬇

Merged Exponents

(a^{m+n})

2 · 2 · 2 · 2 · 2

Exponents and Radicals - Theory & Concepts - Power Function

Detailed guide on the laws of exponents, simplifying radicals, rationalizing denominators, and solving radical equations.

Exponents and Radicals - Theory & Concepts - Radical Equations

Detailed guide on the laws of exponents, simplifying radicals, rationalizing denominators, and solving radical equations.

Radical Equation Visualizer

\sqrt{2x + (4)} = 3

Coefficient a (slope/stretch)2

Constant b (shift)4

Target Value c3

Algebraic Analysis

Real Solution Found:

x = \frac{3^2 - (4)}{2} = 2.50

y = \sqrt{2x + (4)}

y = 3

Linear Equations - Theory & Concepts - Linear Forms

Understanding linear equations, slope, intercepts, parallel and perpendicular lines, and solving inequalities.

Linear Equation Forms Converter

Point-Slope Coordinates

Point 1:

P_1(x_1, y_1)

x_1

-2

y_1

-3

Point 2:

P_2(x_2, y_2)

x_2

y_2

Active Conversions

Point-Slope

y - (-3) = 1.00(x - (-2))

Slope-Intercept

y = 1.00x - 1.00

Standard Form (Ax + By = C)

(5.0)x - (5.0)y = 5.0

Rise / Run5 / 5

Slope (

m

)1.00

y-Intercept (

b

)-1.00

Linear Equations - Theory & Concepts

Understanding linear equations, slope, intercepts, parallel and perpendicular lines, and solving inequalities.

Linear Equation Explorer

Slope m1

Y-intercept b0

Current form

y = x

Converted meaning

y = x

Standard form converts by solving $A x + B y = C$ for $y$ when $B \\neq 0$ . If $B = 0$ , the graph is vertical and the slope is undefined.

State

Line with slope 1 and y-intercept 0

Slope

Intercept cue

y-intercept 0

Linear Equations - Theory & Concepts - Slope Intercept

Understanding linear equations, slope, intercepts, parallel and perpendicular lines, and solving inequalities.

Slope-Intercept Explorer ($y = mx + b$)

y = 1.5 x + 1

Slope m (steepness & direction)1.5

Y-Intercept b (vertical shift)1

Interactive Insights

Slope ($m = 1.5$): Line rises from left to right.

Y-Intercept ($b = 1$): The line intersects the vertical y-axis exactly at point $(0, 1)$.

\text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{3.0}{2} = 1.5

Y-Intercept (0, b)

Run (Horizontal)

Rise (Vertical)

Word Problems - Theory & Concepts - Work Problems

Techniques and strategies for solving common algebraic word problems including age, mixture, work, motion, and clock problems.

Cooperative Work Problems Explorer

\frac{1}{t_A} + \frac{1}{t_B} = \frac{1}{t_{\text{together}}}

Person A Alone Time ($t_A$)4 hours

Person B Alone Time ($t_B$)6 hours

Current Time Explorer0.00 hrs

Theoretical Solution

Adding the rates:

\text{Rate}_A = \frac{1}{4}, \quad \text{Rate}_B = \frac{1}{6}

\text{Combined Rate} = \frac{1}{4} + \frac{1}{6} = \frac{10}{24}

Time working together:

t_{\text{together}} = \frac{4 \times 6}{4 + 6} = 2.40 \text{ hours}

Person A (Alone - takes 4 hrs)0% Complete

Rate: 0.250 job per hour

Person B (Alone - takes 6 hrs)0% Complete

Rate: 0.167 job per hour

A & B Working Together (takes 2.40 hrs)0% Complete

Combined Rate: 0.417 job per hour

Word Problems - Theory & Concepts

Techniques and strategies for solving common algebraic word problems including age, mixture, work, motion, and clock problems.

Distance-Rate-Time Procedure Visualizer

Problem Parameters

Train A Speed

40 mph

Train B Speed

60 mph

A's Head Start

2 hrs

Dist A: 80.0 mi

Dist B: 0.0 mi

Time (t): 0.00 hr

Solution Procedure

1. Define Variables

Let $t$ = time since Train B started (hrs).

Train A's time = $t + 2$ hrs.

Train A's rate = $40$ mph.

Train B's rate = $60$ mph.

2. Build Equations (d = r × t)

Distance A:

d_{A} = 40 (t + 2)

Distance B:

d_{B} = 60 t

3. Solve for Intercept (d_A = d_B)

60 t = 40 (t + 2)

60 t = 40 t + 80

20 t = 80

t = \frac{80}{20}

4. Final Result & Unit Check

Time to catch up: 4.00 hours

Verify distance:

d_B = 60 × 4.00 = 240.00 mi

d_A = 40 × (4.00 + 2) = 240.00 mi

Systems of Equations - Theory & Concepts

Techniques for solving systems of linear equations using graphing, substitution, and elimination methods.

Systems of Linear Equations Explorer

Equation 1 (Blue)

m1 (Slope)1

b1 (Intercept)2

y = x + 2

Equation 2 (Purple)

m2 (Slope)-0.5

b2 (Intercept)-1

y = - 0.5 x - 1

Determinant Test:

D = m_{2} - m_{1} = - 1.50

System Status:

Consistent Independent (Unique Solution)

(x, y) = (- 2.00, 0.00)

Systems of Equations - Theory & Concepts - Systems Algebraic

Techniques for solving systems of linear equations using graphing, substitution, and elimination methods.

Algebraic Systems Solver Lab

Solving Method

Equation 1:

a_1x + b_1y = c_1

2x + 1y = 4

Equation 2:

a_2x + b_2y = c_2

1x - 1y = 2

Algebraic Step Solver

Step 1: Solve Eq 1 for y

y = \frac{4 - 2x}{1}

Step 2: Substitute into Eq 2

1x - 1\left(\frac{4 - 2x}{1}\right) = 2

Step 3: Solve for x and y

x = 2.00, \quad y = 0.00

Determinant (D)-3.0

X Solution2.00

Y Solution0.00

Systems of Equations - Theory & Concepts - Graphical Systems

Techniques for solving systems of linear equations using graphing, substitution, and elimination methods.

System of Linear Equations Explorer

Equation 1 (Blue)

y_{1} = x + 2

Slope

m_1

1.0

Y-Intercept

b_1

Equation 2 (Rose)

y_{2} = - x - 1

Slope

m_2

-1.0

Y-Intercept

b_2

-1

Algebraic Solver

Consistent & Independent

The lines have different slopes, intersecting at exactly one point:

(x, y) = (- 1.50, 0.50)

Equation 1

Equation 2

Intersection

Quadratic Equations - Theory & Concepts

Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula.

Quadratic Equations - Theory & Concepts - Quadratic Vertex

Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula.

Quadratic Vertex Form Explorer

Vertex Equation

y = (x - 0.0)^2 + 0.0

Scale factor (a)1.0

Horizontal shift (h)0.0

Vertical shift (k)0.0

Expanded Standard Form

y = 1.0x^2 + 0.0x + 0.0

Formula:

y = a x^{2} - 2 a h x + (a h^{2} + k)

Vertex (h, k)(0.0, 0.0)

Focus(0.0, 0.25)

Directrixy = -0.25

Quadratic Equations - Theory & Concepts - Discriminant Explorer

Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula.

Discriminant & Quadratic Roots Explorer

y = x^{2} - 2 x - 3

Coefficient a1.0

Coefficient b-2.0

Coefficient c-3

Analysis & Roots

D = b^{2} - 4 a c

D = (- 2)^{2} - 4 (1) (- 3) = 16.0

2 Distinct Real Roots $(D > 0)$

Crosses the x-axis twice:

x_1 = 3.00, \quad x_2 = -1.00

y = ax^2 + bx + c

Real Roots

Vertex

Polynomials - Theory & Concepts

Operations on polynomials, synthetic division, factorization techniques, and finding roots.

Polynomial Explorer (Cubic)

a (End Behavior)1

b (Spread/Shift)0

c (Slope at y-int)-3

d (Y-Intercept)0

f (x) = x^{3} - 3 x

End Behavior:

Down (L) / Up (R)

Approximate Roots:

x ≈ -1.73 x ≈ -0.00 x ≈ 1.73

Turning Points:

(1.00, -2.00)(-1.00, 2.00)

Polynomials - Theory & Concepts - Synthetic Division

Operations on polynomials, synthetic division, factorization techniques, and finding roots.

Synthetic Division Visualizer

Problem

(x^3 + 3x^2 - 4x - 12) \div (x - 2)

Coefficients a, b, c, d[1, 3, -4, -12]

Divisor Root r (divisor is x - r)2

Instruction Step 0

Setup coefficients of the dividend [a, b, c, d] on the top row, and the root r on the left side.

-4

-12

Polynomials - Theory & Concepts - Remainder Theorem

Operations on polynomials, synthetic division, factorization techniques, and finding roots.

Remainder & Factor Theorem Visualizer

Active Polynomial

P(x) = x^3 - 2x^2 - 1x + 2

Divisor (x - k) root k2.0

Lead coefficient a1

Coefficient b-2

Coefficient c-1

Constant d2

Is a Factor!

Since the remainder $P(k) = 0$ , $(x - 2)$ is a perfect factor of the polynomial!

Graph Coordinate(2.0, 0.00)

Remainder R0.00

Rational Expressions - Theory & Concepts - Rational Function

Simplifying, adding, subtracting, multiplying, and dividing rational expressions, and solving rational equations.

Rational Expressions - Theory & Concepts - Asymptote Explorer

Simplifying, adding, subtracting, multiplying, and dividing rational expressions, and solving rational equations.

Rational Function Asymptote Explorer

y = \frac{2x - 1}{x - 2}

a (Num x)2

b (Num const)-1

c (Denom x)1

d (Denom const)-2

Asymptote Formulas

Vertical Asymptote (VA):Set Denominator = 0

1x - 2 = 0 \implies x = 2.00

Horizontal Asymptote (HA):Ratio of Leading Coefficients

y = \frac{a}{c} = \frac{2}{1} = 2.00

Rational Function

Vertical Asymptote

Horizontal Asymptote

Functions and Graphs - Theory & Concepts - Domain Range

Introduction to functions, domain and range, inverse functions, piecewise functions, and symmetry.

Domain and Range Visualizer

Drag the blue point to explore inputs (Domain, x) and their resulting outputs (Range, y) for the function y = -0.5(x - 2)² + 4.

x = 2.0

y = 4.0

Input (Domain)

2.00

Output (Range)

4.00

Functions and Graphs - Theory & Concepts - Function Transformations

Introduction to functions, domain and range, inverse functions, piecewise functions, and symmetry.

Function Transformations

General Form: $g(x) = a \cdot f(b(x - h)) + k$

g(x) = x^2

Scaling & Reflection

a (Vertical Stretch)1

b (Horizontal Stretch)1

Translations

h (Horizontal Shift)0

k (Vertical Shift)0

Original:

f(x) = x^2

Transformed:

g(x)

Logarithms - Theory & Concepts - Logarithm Explorer

Rules of logarithms, natural logs, change of base formula, and solving logarithmic and exponential equations.

Logarithms & Exponentials Explorer

Base (b)2

Domain: $x > 0$

Base: $b > 1$

Asymptote: The log curve approaches $x = 0$ (the y-axis) but never touches it, because $b^y \neq 0$ .

Identity: $b^{\log_b(x)} = x$

Logarithmic Function

y = \log_{2}(x)

Exponential Function

y = 2^{x}

Reflection Line

y = x

Logarithms - Theory & Concepts - Logarithm Rules

Rules of logarithms, natural logs, change of base formula, and solving logarithmic and exponential equations.

Laws of Logarithms Sandbox

Select Logarithm Law

Base (b)10

Value x10

Value y5

Expanded vs Condensed Values

Condensed Left Side

\log_{10}(10 \cdot 5)

1.6990

Expanded Right Side

\log_{10}(10) + \log_{10}(5)

1.6990

Adjust sliders to watch both condensed and expanded mathematical equations update in real-time, verifying logarithmic parity.

Logarithms - Theory & Concepts - Logarithm Scale

Rules of logarithms, natural logs, change of base formula, and solving logarithmic and exponential equations.

Linear vs. Logarithmic Scale

Energy / Intensity Factor (

10^x

)x = 3.0

1100M

Linear Scale

Value:

1.0k

Notice how the linear bar barely moves for small values, then shoots off completely for large ones.

Logarithmic Scale (Base 10)

$\\log_{10}(\\text{Value}):$

3.0

The logarithmic scale grows proportionally to the exponent, compressing huge ranges into manageable numbers.

Order of Magnitude: An increase of $+1$ on a Base-10 logarithmic scale means the underlying value gets 10 times larger. An increase of $+2$ means it gets 100 times larger.

Richter Scale

A magnitude 6.0 earthquake has 10 times the wave amplitude of a 5.0.

Decibels (dB)

Every 10 dB increase represents a 10-fold increase in sound intensity.

pH Scale

A pH of 4 has 10 times the hydrogen ion concentration of pH 5.

Complex Numbers - Theory & Concepts - Complex Arithmetic

Understanding imaginary numbers, complex operations, the complex plane, and De Moivre's Theorem.

Complex Vector Arithmetic Plane

Arithmetic Operation

Vector

z_1 = a + bi

Real part (a)2

Imaginary part (b)3i

Vector

z_2 = c + di

Real part (c)1

Imaginary part (d)-2i

Algebraic Result Evaluation

z_1 + z_2 = (2 + 1) + (3 + -2)i = 3 + 1i

Complex Numbers - Theory & Concepts - Complex Plane

Understanding imaginary numbers, complex operations, the complex plane, and De Moivre's Theorem.

Complex Numbers - Theory & Concepts - De Moivre Theorem

Understanding imaginary numbers, complex operations, the complex plane, and De Moivre's Theorem.

De Moivre's Theorem Explorer

Complex Number z

z = 1.20 \left(\cos 45^\circ + i \sin 45^\circ\right)

z = 1.20 e^{i 0.785}

Magnitude r (Radius)1.20

Angle θ (Degrees)45°

Power nn = 3

Theorem Application

Result:

z^{n}

z^{3} = 1.20^{3} \left(\cos (3 \cdot 45^\circ) + i \sin (3 \cdot 45^\circ)\right)

z^{3} = 1.728 \left(\cos 135^\circ + i \sin 135^\circ\right)

Notice that the magnitude raises geometrically to 1.728, while the angle multiplies linearly to 135° (or 135° coterminal).

z (Original)

z^{n}

(Result)

Intermediates

Matrices and Determinants - Theory & Concepts - Matrix Transform

Comprehensive guide to matrix operations, calculating determinants, finding inverses, and using Cramer's Rule.

2x2 Matrix Transformation

Determinant:1.00

Area Scale:1.00x

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

a (i-hat x)1.0

c (i-hat y)0.0

b (j-hat x)0.0

d (j-hat y)1.0

Matrices and Determinants - Theory & Concepts - Matrix Inverse

Comprehensive guide to matrix operations, calculating determinants, finding inverses, and using Cramer's Rule.

Matrix Inverse & Determinant Visualizer

Matrix A

A = \begin{bmatrix} 2.0 & 1.0 \\ 1.0 & 2.0 \end{bmatrix}

Element a (row 1, col 1)2.0

Element b (row 1, col 2)1.0

Element c (row 2, col 1)1.0

Element d (row 2, col 2)2.0

Inverse Matrix A^-1

A^{-1} = \begin{bmatrix} 0.67 & -0.33 \\ -0.33 & 0.67 \end{bmatrix}

A^{-1} = \frac{1}{\det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Determinant3.00

Area Scale Factor3.00x

Matrices and Determinants - Theory & Concepts - Cramers Rule

Comprehensive guide to matrix operations, calculating determinants, finding inverses, and using Cramer's Rule.

Cramer's Rule 2x2 Matrix Solver

Equation 1

2 x + (1) y = 4

a₁ = 2

b₁ = 1

c₁ = 4

Equation 2

1 x + (- 1) y = 1

a₂ = 1

b₂ = -1

c₂ = 1

Cramer's Step-by-Step Determinants

Main D

D = \begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix} = -3

Dx (Replace x)

D_x = \begin{vmatrix} 4 & 1 \\ 1 & -1 \end{vmatrix} = -5

Dy (Replace y)

D_y = \begin{vmatrix} 2 & 4 \\ 1 & 1 \end{vmatrix} = -2

Eq 1 (Cyan)Eq 2 (Purple)

System Solution

Unique Solution:

x = \frac{D_x}{D} = \frac{-5}{-3} = 1.667

y = \frac{D_y}{D} = \frac{-2}{-3} = 0.667

Conic Sections - Theory & Concepts - Conic General

Equations, properties, and graphs of circles, ellipses, parabolas, and hyperbolas.

Second-Degree Conic & Axis Rotation Explorer

General Equation

(2.0)x^2 + (1.0)xy + (2.0)y^2 = 8

Coefficient A2.0

XY Coefficient B1.0

Coefficient C2.0

Axis Rotation Angle

\theta = 45.0^\circ \quad (0.785\text{ rad})

Eliminates the

xy

term through rotation to produce standard form.

Conic TypeEllipse

Discriminant

(B^2 - 4AC)

-15.0

Conic Sections - Theory & Concepts - Eccentricity Explorer

Equations, properties, and graphs of circles, ellipses, parabolas, and hyperbolas.

Conic Section Focus-Directrix Explorer

Current Conic

Ellipse

\frac{d(P, F)}{d(P, D)} = e = 0.70

Eccentricity (e)0.70

Circle (e=0)Ellipse (0<e<1)Parabola (e=1)Hyperbola (e>1)

Directrix Position (d)x = 3.0

Selected Angle (θ)60°

Focus-Directrix Verification

Focus F: (0.00, 0.00)

Point P: (0.78, 1.35)

Distance PF: 1.556

Distance PD: 2.222

Ratio PF/PD:0.700 (≈ e)

PF (Focus Vector)

PD (Directrix vector)

Sequences and Series - Theory & Concepts - Series Convergence

Arithmetic and geometric progressions, infinite series, summation notation, and the binomial theorem.

Infinite Geometric Series Convergence Lab

Series Equation

S_\infty = \sum_{n=1}^\infty 2(0.50)^{n-1}

First Term (a)2

Common Ratio (r)0.50

Converges! (|r| < 1)

S_\infty = \frac{a}{1 - r} = \frac{2}{1 - 0.50} = 4.00

Partial Sums Progression S_N (N = 1 to 10)

Limit: 4.00

2.00

3.00

3.50

3.75

3.88

3.94

3.97

3.98

3.99

4.00

Partial Sum S_104.00

Infinite Sum S_∞4.00

Sequences and Series - Theory & Concepts - Sequence Series

Arithmetic and geometric progressions, infinite series, summation notation, and the binomial theorem.

Sequence & Series Explorer

Presets:

Starting Value (a₁)2

Common Difference (d)3

Number of Terms (n)10

Term Value (a_{10})

a_{10} = 29

Finite Series Sum (Sₙ)

S_{10} = 155

Loading chart...

Sequences and Series - Theory & Concepts - Binomial Theorem

Arithmetic and geometric progressions, infinite series, summation notation, and the binomial theorem.

Binomial Theorem & Pascal's Triangle

Binomial Power

(a + b)^{4}

Power (n)

n = 4

Selected Term Coefficient index (k)

k = 2

Combinatorial Formulation

\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6

This coefficient represents the term containing $a^{n - k}b^{k}$ in the expanded algebraic series.

Pascal's Triangle

Expansion Series

(a+b)^{4} = a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}

Combinatorics and Probability

Fundamental counting principles, permutations, combinations, and basic probability theory.

Permutations vs. Combinations Explorer

Total Items (

n

Pool of distinct items to choose from

Items Selected (

r

Number of items to select
(Note: r cannot exceed n. Assumes distinct items, no duplicates)

Item Pool (4):

Permutations (Order Matters)

Scenario: Creating a password, or picking 1st/2nd/3rd place.
(Choosing 2 items where order matters)

P(4, 2) = \frac{4!}{(4-2)!} = \frac{24}{2} = 12

12 distinct arrangements

1,21,31,42,12,32,43,13,23,44,14,24,3

Combinations (Order Does NOT Matter)

Scenario: Forming a committee, or drawing a hand of cards.
(Choosing 2items where order doesn't matter)

C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{24}{2 \cdot 2} = 6

6 distinct groups

1,21,31,42,32,43,4

Notice that $P(4, 2)$ is always exactly $2!$ times larger than $C(4, 2)$ , because for every combination (group), there are $2!$ ways to arrange those specific items. ( $2! = 2$ ).

Combinatorics and Probability - Venn Probability

Fundamental counting principles, permutations, combinations, and basic probability theory.

Set Probability & Venn Diagram Simulator

Set Probability Parameters

Probability of A - P(A)0.60

Probability of B - P(B)0.50

Intersection - P(A \cap B)0.30

Max intersection cap: 0.50 (min of P(A), P(B))

Interactive Set Algebra

Union: P(A \cup B)0.80

A only: P(A \cap B')0.30

B only: P(B \cap A')0.20

Complement: P((A \cup B)')0.20

Formula: P(A \cup B) = P(A) + P(B) - P(A \cap B)

Set A only (0.30)

Set B only (0.20)

Intersection (0.30)

Complement (0.20)

Combinatorics and Probability - Probability Tree

Fundamental counting principles, permutations, combinations, and basic probability theory.

Conditional Probability Tree Builder

Probability P(A)0.60

Probability P(B | A)0.80

Probability P(B | A')0.30

Bayes & Total Probability

Total P(B):

P(B) = P(A \cap B) + P(A' \cap B) = 0.480 + 0.120 = 0.600

Bayes P(A | B):

P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.480}{0.600} = 0.800

Event A/A'Event BEvent B' (Not B)