system of linear equations

Examples of system of linear equations in the following topics:

• 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.
  • To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all of the system's equations at the same time.
  • For example, consider the following system of linear equations in two variables:
  • Note that a system of linear equations may contain more than two equations, and more than two variables.
  • Each of these possibilities represents a certain type of system of linear equations in two variables.

• Matrix Equations

  • Matrices can be used to compactly write and work with systems of multiple linear equations.
  • This is very helpful when we start to work with systems of equations.
  • Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants.
  • Using matrix multiplication, we may define a system of equations with the same number of equations as variables as:
  • To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.

• Inconsistent and Dependent Systems

  • Two properties of a linear system are consistency (are there solutions?
  • In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
  • A linear system may behave in any one of three possible ways:
  • The equations of a linear system are independent if none of the equations can be derived algebraically from the others.
  • This is an example of equivalence in a system of linear equations.

• Inconsistent and Dependent Systems in Two Variables

  • For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
  • Recall that a linear system may behave in any one of three possible ways:
  • Also recall that each of these possibilities corresponds to a type of system of linear equations in two variables.
  • We will now focus on identifying dependent and inconsistent systems of linear equations.
  • The equations of a linear system are independent if none of the equations can be derived algebraically from the others.

• Solving Systems Graphically

  • A simple way to solve a system of equations is to look for the intersecting point or points of the equations.
  • The most common ways to solve a system of equations are:
  • This point is considered to be the solution of the system of equations.
  • In a set of linear equations (such as in the image below), there is only one solution.
  • This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system.

• Nonlinear Systems of Equations and Problem-Solving

  • As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables.
  • In a system of equations, two or more relationships are stated among variables.
  • Nonlinear systems of equations, such as conic sections, include at least one equation that is nonlinear.
  • As with linear systems of equations, substitution can be used to solve nonlinear systems for one variable and then the other.
  • Solving nonlinear systems of equations algebraically is similar to doing the same for linear systems of equations.

• Solving Systems of Equations in Three Variables

  • A system of equations in three variables involves two or more equations, each of which contains between one and three variables.
  • This set is often referred to as a system of equations.
  • A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations.
  • This is a set of linear equations, also known as a linear system of equations, in three variables:
  • This images shows a system of three equations in three variables.

• Linear and Quadratic Equations

  • In a set simultaneous equations, or system of equations, multiple equations are given with multiple unknowns.
  • A solution to the system is an assignment of values to all the unknowns so that all of the equations are true.
  • Two kinds of equations are linear and quadratic.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • An example of a graphed linear equation is presented below.

• 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 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.
  • 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 of the form:

• Nonhomogeneous Linear Equations

  • Nonhomogeneous second-order linear equation are of the the form: $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $f(t)$ is nonzero.
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • However, there is a very important property of the linear differential equation, which can be useful in finding solutions.
  • If $y_1(t)$ and $y_2(t)$ are both solutions of the second-order linear differential equation provided above and replicated here:
  • then any arbitrary linear combination of $y_1(t)$ and $y_2(t)$ —that is, $y(x) = c_1y_1(t) + c_2 y_2(t)$ for constants $c_1$ and $c_2$—is also a solution of that differential equation.