dependent variable
(noun)
Affected by a change in input, i.e. it changes depending on the value of the input.
(noun)
An arbitrary output; on the Cartesian plane, the value of
Examples of dependent variable in the following topics:
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Graphical Representations of Functions
- Functions have an independent variable and a dependent variable.
- We assign the value of the function to a variable, in this case $y$, that we call the dependent variable.
- Since the output, or dependent variable is $y$, for function notation often times $f(x)$ is thought of as $y$.
- We say the result is assigned to the dependent variable since it depends on what value we placed into the function.
- Next, substitute these values into the function for $x$, and solve for $f(x)$ (which means the same as the dependent variable $y$): we get the ordered pairs:
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The Cartesian System
- A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.
- The revenue, or output, depends upon the number of cars, or input, that they wash.
- Therefore, the revenue is the dependent variable ($y$), and the number of cars is the independent variable ($x$).
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Formulas and Problem-Solving
- Such an equation may include many variables so long as all are of the first order, and the value of any one variable can be calculated if the values of all the other variables are known.
- Some have more variables than others, but none has a variable of order higher than one.
- The dependent variable (y) represents gratuity (tip) as a function of cost of the bill (x) before gratuity.
- Use a given linear formula to solve for a missing variable
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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.
- Dependent systems have an infinite number of solutions.
- We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions.
- The same is true for dependent systems of equations in three variables.
- Explain what it means, graphically, for systems of equations in three variables to be inconsistent or dependent, as well as how to recognize algebraically when this is the case
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Graphing Equations
- For an equation with two variables, $x$ and $y$, we need a graph with two axes: an $x$-axis and a $y$-axis.
- We'll start by choosing a few $x$-values, plugging them into this equation, and solving for the unknown variable $y$.
- Input values (for the independent variable $x$) from -2 to 2 can be used to obtain output values (the dependent variable $y$) from 5 to 9.
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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.
- Also recall that each of these possibilities corresponds to a type of system of linear equations in two variables.
- They do not add new information about the variables, and the loss of an equation from a dependent system does not change the size of the solution set.
- Subtracting the first equation from the second one, both variables are eliminated and we get $0 = 6$.
- Explain when systems of equations in two variables are inconsistent or dependent both graphically and algebraically.
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Inconsistent and Dependent Systems
- ) and dependency (are the equations linearly independent?
- is a system of three equations in the three variables x, y, z.
- Systems that are not independent are by definition dependent.
- are dependent, because the third equation is the sum of the other two.
- In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation.
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Introduction to Systems of Equations
- 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.
- is a system of three equations in the three variables $x, y, z$.
- Each of these possibilities represents a certain type of system of linear equations in two variables.
- A dependent system has infinitely many solutions.
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Matrix Equations
- Make sure that all of the equations are written in a similar manner, meaning the variables need to all be in the same order.
- Make sure that one side of the equation is only variables and their coefficients, and the other side is just 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.
- If the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.
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Scientific Applications of Quadratic Functions
- Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere.
- Quadratic relationships between variables are commonly found in physical sciences, engineering, and elsewhere.
- If any of the variables $a$, $b$, or $c$ represent functions in themselves, it is often useful to expand the terms, combine like variables, and then re-factor the expression.
- Solving for either charge results in a quadratic equation where the charge is dependent on $r^2$.