Linear Equations

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.