Independent system

Examples of Independent system in the following topics:

• Inconsistent and Dependent Systems

  • ) and dependency (are the equations linearly independent?
  • 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.
  • Systems that are not independent are by definition dependent.
  • A system of equations whose left-hand sides are linearly independent is always consistent.

• Inconsistent and Dependent Systems in Two Variables

  • An independent system of equations has exactly one solution $(x,y)$.
  • The previous modules have discussed how to find the solution for an independent system of equations.
  • The equations of a linear system are independent if none of the equations can be derived algebraically from the others.
  • Systems that are not independent are by definition dependent.
  • A system of equations whose left-hand sides are linearly independent is always consistent.

• Introduction to Systems of Equations

  • 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.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • A solution to the system above is given by
  • An independent system has exactly one solution pair $(x, y)$.
  • An inconsistent system has no solution.

• Inconsistent and Dependent Systems in Three Variables

  • Systems of equations in three variables are either independent, dependent, or inconsistent; each case can be established algebraically and represented graphically.
  • Independent systems have a single solution.
  • Dependent systems have an infinite number of solutions.
  • Inconsistent systems have no solution.
  • We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions.

• The Cartesian System

  • The Cartesian coordinate system is used to specify points on a graph by showing their absolute distances from two axes.
  • Therefore, the revenue is the dependent variable (y) and the number of cars is the independent variable (x).  
  • The four quadrants of a Cartesian coordinate system.
  • The four quadrants of a Cartesian coordinate system.
  • The Cartesian coordinate system with 4 points plotted, including the origin $(0,0)$.

• Matrix Equations

  • Matrices can be used to compactly write and work with systems of multiple linear equations.
  • Matrices can be used to compactly write and work with systems of 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.
  • Thus, we want to solve a system $AX=B$, for $X$.  

• Graphical Representations of Functions

  • Functions have an independent variable and a dependent variable.
  • When we look at a function such as $f(x)=\frac{1}{2}x$, we call the variable that we are changing, in this case $x$, the independent variable.
  • We say that $x$ is independent because we can pick any value for which the function is defined, in this case the set of real numbers $\mathbb{R}$, as inputs into the function.
  • We choose a few values for the independent variable, $x$.  
  • Start by choosing values for the independent variable, $x$.

• Linear Mathematical Models

  • A mathematical model is a description of a system using mathematical concepts and language.
  • Using the slope-intercept form of a linear equation, with the cost labeled $y$ (dependent variable) and the miles labeled $x$ (independent variable):

• 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.
  • 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.
  • We can solve this system algebraically by using equation (1) as a substution.

• Solving Systems of Equations in Three Variables

  • This set is often referred to as a system of equations.
  • The single point where all three planes intersect is the unique solution to the system.
  • This images shows a system of three equations in three variables.
  • The intersecting point (white dot) is the unique solution to this system.
  • Solve a system of equations in three variables, differentiating between systems that have no solutions and ones that have infinitely many solutions