Algebra Studio

Explore algebraic identities, solve linear equations step-by-step, and visualize slope.

Essential Identities

(a+b)² = a² + 2ab + b²
Square of Sum
(a-b)² = a² - 2ab + b²
Square of Diff
a² - b² = (a+b)(a-b)
Diff of Squares

Cubic Identities

(a+b)³ = a³ + 3a²b + 3ab² + b³
Cube of Sum
a³ + b³ = (a+b)(a² - ab + b²)
Sum of Cubes

Exponents

aᵐ · aⁿ = aᵐ⁺ⁿ
Product Rule
(aᵐ)ⁿ = aᵐⁿ
Power Rule
a⁰ = 1
Zero Exponent

Algebra: The Language of Patterns

Algebra is often feared, but it is simply the process of finding the unknown. It gives us a language to describe relationships between numbers using symbols (variables). Whether you are calculating the cost of a trip or determining the trajectory of a rocket, Algebra is the foundation.

This Algebra Studio is designed to demystify the abstract. Do not just memorize formulas—see how equations balance and how lines change slope in real-time.

The Golden Rule: Balance

Imagine an old-fashioned weighing scale. The equals sign (=) represents the center of that scale. The most important rule in algebra is simple: Whatever you do to one side, you must do to the other.

2x + 5 = 15
- 5      - 5 (Subtract 5 from BOTH sides)
----------------
2x       = 10

If you only subtract 5 from the left, the scale tips. Our Linear Solver tool above demonstrates this step-by-step for any equation you enter.

Variables are Containers

What is x? Beginners often get confused by letters in math. Think of x not as a letter, but as an empty box or a container.

When we say `2x + 10 = 20`, we are saying: "Two identical boxes plus 10 pounds equals 20 pounds. How much is in one box?"
By removing the 10 pounds from the total structure, we see the two boxes weigh 10 pounds together. So, one box must hold 5 pounds.

Slope: Rise over Run

In the equation y = mx + b, the letter m represents the Slope (steepness). It is often described as "Rise over Run"—how much you go UP for every step you go RIGHT.

  • High m (e.g., 5): A very steep hill.
  • Low m (e.g., 0.5): A gentle ramp.
  • Negative m: Going downhill.
  • Zero m: A perfectly flat floor.

Why Learn Identities?

Formulas like (a+b)² = a² + 2ab + b² are called Identities. They are true for any value of 'a' and 'b'.

Why memorize them? Speed. In higher math (Calculus), you will frequently need to expand or factor expressions instantly. Recognizing that `x² - 9` factors into `(x+3)(x-3)` immediately allows you to solve limits and derivatives that would otherwise be stuck.

Frequently Asked Questions

What is the difference between an Expression and an Equation?

An expression is a phrase (like '2x + 5'), while an equation is a sentence asserting equality (like '2x + 5 = 15'). An equation can be solved; an expression can only be simplified.

How do I isolate a variable?

To isolate x, perform the inverse (opposite) operation of whatever is happening to x. If x is multiplied by 2, divide by 2. If 5 is added to x, subtract 5. Always start with addition/subtraction before multiplication/division (reverse PEMDAS).

What does FOIL mean?

FOIL stands for First, Outer, Inner, Last. It is a mnemonic for multiplying two binomials, like (x+2)(x+3). First (xx), Outer (x3), Inner (2x), Last (23).

Why can't I divide by zero?

Division is asking 'how many times does this fit?'. Asking how many 0s fit into 10 makes no sense—you can add infinite zeros and never reach 10. In the context of slope, dividing by zero represents a vertical line, which has an 'undefined' slope.

What is the degree of a polynomial?

The degree is the highest exponent (power) on the variable. a linear equation (x¹) has degree 1. A quadratic (x²) has degree 2. The degree tells you the maximum number of times the graph can cross the x-axis.

What if there are variables on both sides?

If you have 2x + 5 = x + 10, the strategy is to 'gather' the variables. Subtract the smaller x term from both sides (subtract x here) to get x + 5 = 10. Then solve normally.

What is the Y-intercept?

The y-intercept (b) is where the line crosses the vertical y-axis. It is the starting value when x = 0. In real life, this often represents a flat fee or starting balance.

Does (a+b)² equal a² + b²?

NO! This is the most common mistake in algebra. (a+b)² means (a+b)(a+b). When you expand it properly, you get a² + ab + ab + b², which simplifies to a² + 2ab + b². You cannot just distribute the exponent.

What is a coefficient?

A coefficient is the number multiplied by a variable. In the term 5x, 5 is the coefficient. It tells you 'how many' x's you have.

How do I check my answer?

Once you find a value for x (e.g., x=5), plug it back into the ORIGINAL equation. If 2(5) + 5 = 15 becomes 10 + 5 = 15 (which is true), then your answer is correct. Checking is the only way to be 100% sure.