Equation (algebra)
An algebraic equation is a mathematical statement asserting that two expressions are equal, connected by an equals sign (=). Solving an equation means finding the value(s) of the variable(s) that make the statement true, called the solution or root. Equations are fundamental to all branches of mathematics and science, providing a precise language for describing quantitative relationships.
Key Formula
f(x) = g(x)
LaTeX: f(x) = g(x)
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Left-hand side expression | dimensionless |
| g(x) | Right-hand side expression | dimensionless |
| x | Variable to be solved | dimensionless |
Worked Example
Problem
Solve the equation 3x − 8 = 2x + 5.
Solution
Step 1: Collect variable terms on one side. 3x − 2x − 8 = 5 Step 2: Simplify the left side. x − 8 = 5 Step 3: Isolate x by adding 8 to both sides. x = 5 + 8 x = 13 Step 4: Verify by substituting back. 3(13) − 8 = 39 − 8 = 31 2(13) + 5 = 26 + 5 = 31 ✓
Answer
x = 13
Types of Algebraic Equations by Degree
| Type | Degree | Standard Form | Number of Solutions |
|---|---|---|---|
| Linear | 1 | ax + b = 0 | 1 real solution |
| Quadratic | 2 | ax² + bx + c = 0 | Up to 2 solutions |
| Cubic | 3 | ax³ + bx² + cx + d = 0 | Up to 3 solutions |
| Quartic | 4 | ax⁴ + … = 0 | Up to 4 solutions |
| Polynomial degree n | n | aₙxⁿ + … = 0 | Up to n solutions |
Interactive Tools
Wolfram Alpha – Equation Solver
Solve any algebraic equation with step-by-step working shown.
Open ToolDesmos Graphing Calculator
Graph both sides of an equation to visualise where solutions occur.
Open ToolKhan Academy – One-variable Equations
Step-by-step video lessons on setting up and solving equations.
Open Tool
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Related Terms
Linear Equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power, producing a straight-line graph when plotted. The standard form of a linear equation in one variable is ax + b = 0, while in two variables it is ax + by = c. Linear equations are foundational in algebra and appear throughout science, economics, and engineering for modelling proportional relationships.
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and it can have two, one, or no real solutions depending on the value of the discriminant (b² − 4ac). Quadratic equations model projectile motion, area problems, and many optimisation scenarios in physics and engineering.
System of Equations
A system of equations is a set of two or more equations containing the same variables, where the solution must satisfy all equations simultaneously. The solution can be found using substitution, elimination, or matrix methods (such as Gaussian elimination), and the system can have a unique solution, infinitely many solutions, or no solution depending on how the equations relate geometrically. Systems of equations are widely applied in engineering, economics, and science to model situations with multiple constraints.
From Latin "aequatio" meaning "an equal distribution", derived from "aequare" (to make equal) and "aequus" (equal or level). The equals sign (=) was invented by Welsh mathematician Robert Recorde in 1557 in his book "The Whetstone of Witte", who chose two parallel lines because "no two things can be more equal".