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

Value of

a

Value of

b

-2

Commutative Property of Addition:

a + b = b + a

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

1 = 1

Visualize $a + b$ and $b + a$ reaching the same sum.

Exponents and Radicals - Theory & Concepts

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

Power Function Explorer

Exponent (n)2

Try integers (2, 3), negative (-1, -2), and fractions (0.5).

y = x^{2}

Linear Equations - Theory & Concepts

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

Linear Equation Explorer: $y = mx + b$

Slope (m)1

Controls steepness and direction.

Y-Intercept (b)0

Where the line crosses the Y-axis.

y = x

Word Problems - Theory & Concepts

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

Distance-Rate-Time Visualizer (Catch Up Problem)

Train A Speed (mph): 40

Train B Speed (mph): 60

Train A Head Start (hrs): 2

Dist A: 80.0 miles

Dist B: 0.0 miles

Live Statistics

Time passed since B started: 0.00 hours
Total time A traveled: 2.00 hours
Distance between them: 80.0 miles

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 = -0.5x - 1

Solution:

x = -2.00

y = 0.00

Intersection Point

Quadratic Equations - Theory & Concepts

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

Quadratic Explorer: $y = ax^2 + bx + c$

a (Quadratic)1

b (Linear)0

c (Constant)0

y = x^2

Vertex: (0.00, 0.00)

Polynomials - Theory & Concepts

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

Polynomial Explorer (Cubic)

a (x³)1

b (x²)0

c (x)-3

d (const)0

f(x) = x^3 - 3x

Approximate Roots:

x ≈ -1.73x ≈ -0.00x ≈ 1.73

Rational Expressions - Theory & Concepts

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

Functions and Graphs - Theory & Concepts

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

Logarithms - Theory & Concepts

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.

Example Context: If $1$ is a microscopic tremor, $10^8$ is a massive earthquake. The Richter Scale uses logarithms so we can say "Magnitude 8" instead of "Intensity 100,000,000".

Complex Numbers - Theory & Concepts

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

Complex Plane (Argand Diagram)

r = 5.00|θ = 53.1°

Rectangular Form

z = 3 + 4i

Real Part (a)3

Imaginary Part (b)4

Polar Form

z = 5.00(\cos 53.1^\circ + i\sin 53.1^\circ)

Matrices and Determinants - Theory & Concepts

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

2x2 Matrix Transformation

Area:1.00

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

Conic Sections - Theory & Concepts

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

Sequences and Series - Theory & Concepts

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

Sequence & Series Explorer

Starting Value (a₁)2

Common Difference (d)3

Number of Terms (n)10

Term Value (aₙ)

a_{10} = 29

Series Sum (Sₙ)

S_{10} = 155

Loading chart...

Combinatorics and Probability - Theory & Concepts

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

Item Pool (4):

Permutations (Order Matters)

Example: Passwords, Rankings (1st, 2nd, 3rd)

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

12 distinct arrangements

Combinations (Order Does NOT Matter)

Example: Committees, Hands of cards

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

6 distinct groups

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. ( $2! = 2$ ).