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.
Worked Example
Problem
Solve the system: 2x + 3y = 12 and 4x − y = 5 using the elimination method.
Solution
Step 1: Multiply the second equation by 3 to match the y-coefficient. 3(4x − y) = 3(5) 12x − 3y = 15 Step 2: Add the two equations to eliminate y. (2x + 3y) + (12x − 3y) = 12 + 15 14x = 27 x = 27/14 Step 3: Substitute x = 27/14 back into 2x + 3y = 12. 2(27/14) + 3y = 12 27/7 + 3y = 12 3y = 12 − 27/7 = 84/7 − 27/7 = 57/7 y = 19/7 Step 4: Verify in 4x − y = 5. 4(27/14) − 19/7 = 108/14 − 38/14 = 70/14 = 5 ✓
Answer
x = 27/14, y = 19/7
Methods for Solving Systems of Linear Equations
| Method | Description | Best For | Limitation |
|---|---|---|---|
| Substitution | Solve one equation for a variable, substitute into other | Small systems, one easy-to-isolate variable | Can produce messy fractions |
| Elimination (addition) | Multiply equations to cancel one variable | Systems where coefficients match easily | Requires careful multiplier choice |
| Graphing | Find intersection point of lines | Visualising solutions | Imprecise for non-integer solutions |
| Matrix (Gaussian) | Row-reduce augmented matrix | Large systems with many variables | Requires knowledge of matrices |
| Cramer's Rule | Use determinants | Systems with unique solutions | Inefficient for large systems |
Interactive Tools
Wolfram Alpha – System Solver
Solve systems of two or more equations with full step-by-step solutions.
Open ToolDesmos Graphing Calculator
Graph multiple equations simultaneously and identify intersection points.
Open ToolKhan Academy – Systems of Equations
Comprehensive coverage of all methods for solving linear systems.
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.
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.
Variable (algebra)
A variable is a symbol, typically a letter such as x, y, or n, that represents an unknown or changing quantity in a mathematical expression or equation. Variables allow mathematicians to write general rules and relationships that apply to many specific cases at once. In algebra, manipulating variables to solve for unknowns or express patterns is the central skill.
From Greek "systema" meaning "a whole compounded of several parts", derived from "synistanai" (to bring together). The word "equation" comes from Latin "aequatio" (a making equal). The systematic study of simultaneous equations dates to ancient China (the "Nine Chapters on the Mathematical Art", circa 200 BCE) and was developed further by European mathematicians in the 17th–19th centuries.