Examples of dependent system in the following topics:
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- For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
- An inconsistent system has no solution, and a dependent system has an infinite number of solutions.
- We will now focus on identifying dependent and inconsistent systems of linear equations.
- Systems that are not independent are by definition
dependent.
- We can apply the substitution or elimination methods for solving systems of equations to identify dependent systems.
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- 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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- ) and dependency (are the equations linearly independent?
- 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.
- The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are not linearly independent, i.e. are dependent.
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- A system of equations consists of two or more equations with two or more variables, where any solution must satisfy all of the equations in the system at the same time.
- 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.
- A solution to the system above is given by
- An inconsistent system has no solution.
- A dependent system
has infinitely many solutions.
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- The revenue, or output, depends upon the number of cars, or input, that have their cars washed.
- 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 $(0,0)$.
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- 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.
- Thus, we want to solve a system $AX=B$, for $X$.
- No, 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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- Nonlinear systems of equations can be used to solve complex problems involving multiple known relationships.
- Three or more signals reduce the solution of the system to a single coordinate point.
- The kinetic energy of the objects depends on the speed squared, and the momentum depends on the speed directly.
- In addition to practical scenarios like the above, nonlinear systems can be used in abstract problems.
- Extend the ideas behind nonlinear systems of equations to real world applications
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- Second, shade either above or below the line, depending on if $y$ is greater or less than $mx+b$.
- These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
- This also means that if there are inequalities that don't overlap, then there is no solution to the system.
- The brown overlapped shaded area is the final solution to the system of linear inequalities because it is comprised of all possible solutions to $y<-\frac{1}{2}x+1$ (the dotted red line and red area below the line) and $y\geq x-2$ (the solid green line and the green area above the line).
- The origin is a solution to the system, but the point $(3,0)$ is not.
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- The angle between the ground plane and the sunlight cone depends on where you are on the Earth, and the axial tilt of Earth, which changes seasonally.
- Depending on the orbital properties, including size and shape (eccentricity), this orbit can be any of the four conic sections.
- A parabolic trajectory does have the particle escaping the system.
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- The acidity depends on the hydrogen ion concentration in the liquid (in moles per liter) written as [H+].
- The entropy (S) of a system can be calculated from the natural logarithm of the number of possible microstates (W) the system can adopt:
- In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones.