Equality is a statement
The equals sign says that the expression on the left and the expression on the right represent the same value.
ALGEBRA · FOUNDATIONAL TOPIC
Equations
An equation states that two mathematical expressions have the same value. Solving it means finding every value of the unknown that makes that statement true.
2x + 3 = 11
preserve equality on both sidesx = 4
START WITH THE IDEA
What does solving an equation mean?
Three ideas make nearly every algebraic solving method easier to understand.
Solutions make it true
A solution is a value that satisfies the original equation. The collection of all such values is its solution set.
Transform, but preserve
We replace an equation by an equivalent one using operations that preserve exactly the same solutions.
CORE PRINCIPLE
Do the same valid operation to both sides.
Add or subtract the same quantity on both sides.
Multiply or divide both sides by the same non-zero quantity.
Simplify carefully and keep track of restrictions when operations may not be reversible.
Check the result in the original equation, especially after squaring, taking roots or clearing variable denominators.
EQUATION TYPES
Choose the structure you are solving.
The form of an equation determines which methods are reliable and efficient.
a = b
Equation fundamentals
Equality, solution sets, equivalent transformations and domain restrictions.
Learn the foundations →ax + b = 0
Linear equations
One unknown, first degree. Solve algebraically and interpret the answer graphically.
Study linear equations →ax² + bx + c = 0
Quadratic equations
Factoring, completing the square, the quadratic formula and discriminant.
Study quadratics →ax³ + … = 0
Cubic equations
Factorisation, roots and transformations for third-degree polynomial equations.
Explore cubic equations →P(x) = 0
Polynomial equations
Root structure and practical strategies for equations of higher degree.
Explore polynomials →f(x) = g(x)
Special equation types
Radical, rational, exponential and other equations requiring additional care.
Explore special types →RECOMMENDED PATH
Build understanding in the right order.
Equation fundamentals→Linear equations→Quadratic equations→Systems→Advanced equations