simultaneous equations

(noun)

finite sets of equations whose common solutions are looked for

Related Terms

Examples of simultaneous equations in the following topics:

  • Parametric Equations

    • Parametric equations are a set of equations in which the coordinates (e.g., $x$ and $y$) are expressed in terms of a single third parameter.
    • Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations.
    • If one of these equations can be solved for $t$, the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ only.
    • In some cases there is no single equation in closed form that is equivalent to the parametric equations.
    • One example of a sketch defined by parametric equations.

  • 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:

  • Graphing on Computers and Calculators

    • A graphing calculator (see ) typically refers to a class of handheld scientific calculators that are capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables.

  • Solving Differential Equations

    • Differential equations are solved by finding the function for which the equation holds true.
    • A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.
    • As you can see, such an equation relates a function $f(x)$ to its derivative.
    • Solving the differential equation means solving for the function $f(x)$.
    • The "order" of a differential equation depends on the derivative of the highest order in the equation.

  • Nonhomogeneous Linear Equations

    • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
    • Otherwise, the equations are called nonhomogeneous equations.
    • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
    • This can be confirmed by substituting $y(x) = c_1y_1(t) + c_2 y_2(t)$ into the equation and using the fact that both $y_1(t)$ and $y_2(t)$ are solutions of the equation.
    • Identify when a second-order linear differential equation can be solved analytically

  • Models Using Differential Equations

    • Differential equations can be used to model a variety of physical systems.
    • Differential equations are very important in the mathematical modeling of physical systems.
    • The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application.
    • Conduction of heat is governed by another second-order partial differential equation, the heat equation .
    • Give examples of systems that can be modeled with differential equations

  • Separable Equations

    • Separable differential equations are equations wherein the variables can be separated.
    • One of these forms is separable equations.
    • A separable equation is a differential equation of the following form:
    • The original equation is separable if this differential equation can be expressed as:
    • Integrating such an equation yields:

  • Second-Order Linear Equations

    • For a function dependent on time, we may write the equation more expressly as $L y(t) = f(t)$ and, even more precisely, by bracketing $L [y(t)] = f(t)$.
    • It is convenient to rewrite this equation in an operator form:
    • 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.

  • Predator-Prey Systems

    • The relationship between predators and their prey can be modeled by a set of differential equations.
    • The predator–prey equations are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey.
    • As differential equations are used, the solution is deterministic and continuous.
    • The solutions to the equations are periodic.
    • Identify type of the equations used to model the predator-prey systems

  • Curve Sketching

    • Curve sketching is used to produce a rough idea of overall shape of a curve given its equation without computing a detailed plot.
    • The $x$-intercepts are found by setting $y$ equal to $0$ in the equation of the curve and solving for $x$.
    • Similarly, the y intercepts are found by setting $x$ equal to $0$ in the equation of the curve and solving for $y$.
    • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
    • For algebraic curves, this can be done by removing all but the terms of lowest order from the equation and solving.