linear equation

(noun)

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

Related Terms

  • A
  • univariate
  • first order
  • first power
  • general (standard) form
  • real numbers
  • rate of change
  • quadratic equation
  • slope
  • inverse matrix
  • inequality
  • constraint

(noun)

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

Related Terms

  • A
  • univariate
  • first order
  • first power
  • general (standard) form
  • real numbers
  • rate of change
  • quadratic equation
  • slope
  • inverse matrix
  • inequality
  • constraint

Examples of linear equation in the following topics:

  • 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:
    • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
    • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
    • Linear differential equations are of the form:
  • Linear Equations and Their Applications

    • Linear equations are those with one or more variables of the first order.
    • There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
    • Linear equations can therefore be expressed in general (standard) form as:
    • For example,imagine these linear equations represent the trajectories of two vehicles.
    • Imagine these linear equations represent the trajectories of two vehicles.
  • Introduction to Systems of Equations

    • A system of linear equations consists of two or more linear equations made up of two or more variables, such that all equations in the system are considered simultaneously.
    • For example, consider the following system of linear equations in two variables:
    • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
    • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
    • Note that a system of linear equations may contain more than two equations, and more than two variables.
  • Nonhomogeneous Linear Equations

    • In the previous atom, we learned that a second-order linear differential equation has the form:
    • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
    • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
    • Phenomena such as heat transfer can be described using nonhomogeneous second-order linear differential equations.
    • Identify when a second-order linear differential equation can be solved analytically
  • Linear Equations in Standard Form

    • A linear equation written in standard form makes it easy to calculate the zero, or $x$-intercept, of the equation.
    • Standard form is another way of arranging a linear equation.
    • In the standard form, a linear equation is written as:
    • However, the zero of the equation is not immediately obvious when the linear equation is in this form.
    • Convert linear equations to standard form and explain why it is useful to do so
  • Linear Equations

    • Linear regression for two variables is based on a linear equation with one independent variable.
    • The graph of a linear equation of the form y = a + bx is a straight line.
    • Any line that is not vertical can be described by this equation.
    • Linear equations of this form occur in applications of life sciences, social sciences, psychology, business, economics, physical sciences, mathematics, and other areas.
    • Find the equation that expresses the total cost in terms of the number of hours required to finish the word processing job.
  • Linear and Quadratic Equations

    • Two kinds of equations are linear and quadratic.
    • Linear equations can have one or more variables.
    • Linear equations do not include exponents.
    • An example of a graphed linear equation is presented below.
    • (If $a=0$, the equation is a linear equation.)
  • Inconsistent and Dependent Systems

    • In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
    • The equations of a linear system are independent if none of the equations can be derived algebraically from the others.
    • For linear equations, logical independence is the same as linear independence.
    • This is an example of equivalence in a system of linear equations.
    • It is possible for three linear equations to be inconsistent, even though any two of them are consistent together.
  • Second-Order Linear Equations

    • A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
    • 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 the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
    • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
    • (Otherwise, the equations are called nonhomogeneous equations.)
    • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.
  • Linear and Quadratic Functions

    • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
    • Linear functions may be confused with affine functions.
    • Linear functions form the basis of linear algebra.
    • If the quadratic function is set equal to zero, then the result is a quadratic equation.
    • The solutions to the equation are called the roots of the equation.
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