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.
Key Formula
y = mx + c
LaTeX: y = mx + c
| Symbol | Meaning | Unit |
|---|---|---|
| y | Dependent variable (output) | dimensionless |
| m | Slope (gradient) of the line | dimensionless |
| x | Independent variable (input) | dimensionless |
| c | y-intercept (value of y when x = 0) | dimensionless |
Worked Example
Problem
Find the equation of the line passing through (2, 5) with slope 3, then find y when x = 7.
Solution
Step 1: Use slope-intercept form y = mx + c with m = 3. y = 3x + c Step 2: Substitute the known point (2, 5) to find c. 5 = 3(2) + c 5 = 6 + c c = −1 Step 3: Write the equation. y = 3x − 1 Step 4: Evaluate at x = 7. y = 3(7) − 1 = 21 − 1 = 20
Answer
y = 3x − 1; when x = 7, y = 20
Forms of Linear Equations in Two Variables
| Form | Equation | Parameters | Best Used When |
|---|---|---|---|
| Slope-intercept | y = mx + c | m = slope, c = y-intercept | Graphing or reading off slope |
| Standard form | ax + by = c | a, b, c are integers | Systems of equations |
| Point-slope | y − y₁ = m(x − x₁) | m = slope, (x₁,y₁) = known point | Writing from a point and slope |
| Intercept form | x/a + y/b = 1 | a = x-intercept, b = y-intercept | When both intercepts are known |
Interactive Tools
Desmos Graphing Calculator
Plot linear equations and experiment with slope and intercept sliders.
Open ToolGeoGebra – Linear Functions
Interactive graphing tool to explore linear equations in multiple forms.
Open ToolKhan Academy – Linear Equations
Comprehensive lessons on graphing and solving linear equations.
Open Tool
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Related Terms
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.
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.
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.
From Latin "linearis" meaning "belonging to a line", derived from "linea" (a line or string). The term reflects the geometric fact that solutions to a linear equation in two variables form a straight line when graphed on a coordinate plane.