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Suppose we have two linear equations in two unknowns. If the equations are plotted on a rectangular grid, the graph will fit one of these scenarios:
1) The two lines cross each other (intersect).
2) The two lines don't cross - they are parallel lines
3) The two lines fall on top of each other - they're really the same line.
In case 3) the two lines are dependent - one can be changed into the other. In cases 1) and 2) we say the lines are independent.
If the pair of equations has a solution (one or more points in common) we say they are consistent ... cases 1) and 3). In case 2) the system is inconsistent; there is no solution.
To summarize:
1) Intersecting lines are consistent and independent.
2) Parallel lines are inconsistent and independent.
3) Coincident ["happen together"] lines are consistent and dependent.
*** A second order linear system CANNOT be both dependent and inconsistent.
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How can you identify a dependent or inconsistent system by looking at an augmented matrix in reduced row-echelon form?
And your question is......................?
What do both dependent and inconsistent systems have in common?
no solution
What describes a system of equations that has no solution?
Inconsistent.
When will a system of linear equations be inconsistent?
When the matrix of coefficients is singular.
What is consistent and dependent?
The terms consistent and dependent are two ways to describe a system of linear equations. A system of linear equations is dependent if you can algebraically derive one of the equations from one or more of the other equations. A system of linear equations is consistent if they have a common solution.An example of a dependent system of linear equations:2x + 4y = 84x + 8y = 16Solve the first equation for x:x = 4 - 2yPlug that value of x into the second equation:16 - 8y + 8y = 16, which gives 16 = 16.No new information was gained from the second equation, because we already knew 16 = 16, so these two equations are dependent.An example of an inconsistent system of linear equations:Because consistency is boring.2x + 4y = 84x + 8y = 15Solve the first equation for x:x = 4 - 2yPlug that value of x into the second equation:16 - 8y + 8y = 15, which gives 16 = 15.This is a contradiction, because 16 doesn't equal 15. Therefore this system has no solution and is inconsistent.
