variable

Examples of variable in the following topics:

• Introduction to Variables

  • Variables are used in mathematics to denote arbitrary or unknown numbers.
  • Variables can represent numbers whose values are not yet known.
  • Variables may describe some mathematical properties.
  • To distinguish among the different variables, $x$ is called an unknown, and the variables that are multiplied by $x$ are called coefficients.
  • Therefore, a term may simply be a constant or a variable, or it may include both a coefficient and an unknown variable.

• What is an Equation?

  • Equations with variables have solutions, or values for the variables that make the statements true.
  • In many cases, an equation contains one or more variables.
  • It is possible for equations to have more than one variable.
  • For example, $x + y + 7 = 13$ is an equation in two variables.
  • When an equation contains a variable such as $x$, this variable is considered an unknown value.

• Equations in Two Variables

  • Equations in two variables represent the relationship between two variables and have a series of solutions.
  • Equations can have multiple variables; such equations express the relationship between its variables.
  • Equations in two variables often express a relationship between the variables $x$ and $y$, which correspond to Cartesian coordinates.
  • Equations in two variables have not only one solution, but a series of solutions that satisfy the equation for both variables.
  • For a given equation in two variables, choosing a value for one variable dictates what the value of the other variable will be.

• The Elimination Method

  • The elimination method is used to eliminate a variable in order to more simply solve for the remaining variable(s) in a system of equations.
  • The elimination method for solving systems of equations, also known as elimination by addition, is a way to eliminate one of the variables in the system in order to more simply evaluate the remaining variable.
  • Once the values for the remaining variables have been found successfully, they are substituted into the original equation in order to find the correct value for the other variable.
  • First, line up the variables so that the equations can be easily added together in a later step:
  • If not, multiply one equation by a number that allow the variables to cancel out.

• 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.
  • In a system of equations in three variables, you can have one or more equations, each of which may contain one or more of the three variables, usually x, y, and z.
  • The substitution method of solving a system of equations in three variables involves identifying an equation that can be easily by written with a single variable as the subject (by solving the equation for that variable).
  • Next, substitute that expression where that variable appears in the other two equations, thereby obtaining a smaller system with fewer variables.
  • As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable.

• Direct and Inverse Variation

  • Two variables in direct variation have a linear relationship, while variables in inverse variation do not.
  • Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other.
  • The two variables may be considered directly proportional.
  • Doing so, the variables would abide by the relationship:
  • In fact, two variables are said to be inversely proportional when an operation of change is performed on one variable and the opposite happens to the other.

• Direct Variation

  • When two variables change proportionally to each other, they are said to be in direct variation.
  • Knowing that the relationship between two variables is constant, we can show their relationship as :
  • Doing so, the variables would abide by the relationship:
  • Any augmentation of one variable would lead to an equal augmentation of the other.
  • The line y=kx is an example of direct variation between variables x and y.

• The Substitution Method

  • The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation.
  • When the resulting simplified equation has only one variable to work with, the equation becomes solvable.
  • In the first equation, solve for one of the variables in terms of the others.
  • Note that now this equation only has one variable (y).
  • Now that we know the value of y, we can use it to find the value of the other variable, x.

• Combined Variation

  • Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other.
  • The two variables may be considered directly proportional.
  • Two variables are said to be in inverse variation, or are inversely proportional, when an operation of change is performed on one variable and the opposite happens to the other.
  • To have variables that are in combined variation, the equation must have variables that are in both direct and inverse variation, as shown in the example below.
  • Solving for P, we can determine the variation of the variables.

• 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 assign the value of the function to a variable, in this case $y$, that we call the dependent variable.  
  • We choose a few values for the independent variable, $x$.  
  • Start by choosing values for the independent variable, $x$.