Calculus
Textbooks
Boundless Calculus
Differential Equations, Parametric Equations, and Sequences and Series
Differential Equations
Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series Differential Equations
Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 9
Created by Boundless

Linear Equations

Linear equations are equations of a single variable.

Learning Objective

  • Write an expression for a linear differential equation


Key Points

    • Linear equations involve a single variable and an arbitrary number of constants.
    • Linear equations are so-called because their most basic form is described by a line on a graph.
    • Linear differential equations are differential equations which involve a single variable and its derivative.

Terms

  • simultaneous equations

    finite sets of equations whose common solutions are looked for

  • linear equation

    a polynomial equation of the first degree (such as $x = 2y - 7$)

  • differential equation

    an equation involving the derivatives of a function


Full Text

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.

A common form of a linear equation in the two variables $x$ and $y$ is:

$y=mx+b$

where $m$ and $b$ designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.

Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than $1$) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin x$ are nonlinear.

Linear equations can be written in parametric form:

$x=Tt+U \\ y= Vt+W$

The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values: 

  • slope ($m$):  $\displaystyle{\frac{V }{ T}}$
  • x-intercept: $\displaystyle{\frac{VU−WT}{V}}$
  • y-intercept: $\displaystyle{\frac{ WT−VU}{ T}}$

 

Linear differential equations are differential equations that have solutions which can be added together to form other solutions. They can be ordinary or partial. Linear differential equations are of the form:

$Ly=f$

where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and $f$ is a given function of the same nature as y (called the source term). For a function dependent on time, we may write the equation more expressly as:

$Ly(t)=f(t)$

Linear equations

Graphical example of linear equations.

[ edit ]
Edit this content
Prev Concept
Logistic Equations and Population Grown
Predator-Prey Systems
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.