linear equation

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.