· determine whether a mechanism of linear equations is continuous or inconsistent.
You are watching: Which system of equations is consistent and dependent?
· determine whether a mechanism of straight equations is dependency or independent.
· determine whether an ordered pair is a equipment of a device of equations.
· fix application difficulties by graphing a mechanism of equations.
Recall the a direct equation graphs as a line, which indicates that every one of the points on the heat are solutions to that direct equation. There space an infinite number of solutions. If you have actually a system of linear equations, the systems for the mechanism is the value that makes all of the equations true. For 2 variables and also two equations, this is the point where the two graphs intersect. The works with of this point will be the equipment for the 2 variables in the two equations.
The solution for a mechanism of equations is the value or worths that are true for all equations in the system. The graphs that equations in ~ a system deserve to tell friend how numerous solutions exist for that system. Look in ~ the photos below. Each shows two lines that consist of a mechanism of equations.
One Solution  No Solutions  Infinite Solutions 
If the graphs of the equations intersect, then there is one solution that is true because that both equations.  If the graphs that the equations do not intersect (for example, if they room parallel), climate there room no remedies that space true because that both equations.  If the graphs that the equations are the same, climate there are an infinite variety of solutions that are true for both equations. 
When the currently intersect, the point of intersection is the only point that the 2 graphs have actually in common. Therefore the collaborates of that suggest are the systems for the 2 variables used in the equations. Once the lines room parallel, there space no solutions, and sometimes the 2 equations will graph together the very same line, in which situation we have actually an infinite variety of solutions.
Some unique terms are sometimes used to explain these kinds of systems.
The complying with terms refer to how countless solutions the device has.
o as soon as a system has actually one equipment (the graphs that the equations crossing once), the device is a consistent mechanism of linear equations and the equations space independent.
o once a system has actually no equipment (the graphs that the equations don’t crossing at all), the device is one inconsistent system of direct equations and also the equations space independent.
o If the lines space the very same (the graphs intersect at every points), the system is a continual system of straight equations and also the equations space dependent. That is, any type of solution that one equation must also be a systems of the other, so the equations depend on each other.
The adhering to terms describe whether the device has any solutions in ~ all.
o The device is a continuous system of straight equations when it has solutions.
o The system is an inconsistent mechanism of straight equations as soon as it has actually no solutions.
We can summarize this together follows: o A system with one or an ext solutions is consistent. o A mechanism with no solutions is inconsistent. o If the lines are different, the equations room independent straight equations. o If the lines room the same, the equations space dependent linear equations. 
Example  
Problem  Using the graph the y = x and x + 2y = 6, presented below, determine how countless solutions the mechanism has. Climate classify the mechanism as consistent or inconsistent and also the equations together dependent or independent.  
The lines intersect at one point. For this reason the 2 lines have actually only one allude in common, over there is just one solution to the system. Because the lines space not the very same the equations are independent. Because there is just one solution, this mechanism is consistent.  
Answer  The mechanism is consistent and the equations room independent.  
Advanced Example  
Problem  Using the graph of y = 3.5x + 0.25 and also 14x – 4y = 4.5, displayed below, determine how many solutions the device has. Then classify the system as continual or inconsistent and the equations as dependent or independent.  
The lines space parallel, an interpretation they perform not intersect. There space no services to the system. The lines space not the same, the equations space independent. There are no solutions. Therefore, this device is inconsistent.  
Answer  The system is inconsistent and also the equations room independent.  
Advanced Question Which of the complying with represents dependent equations and consistent systems? A) B) C) D) A) Incorrect. The 2 lines in this system have actually the exact same slope, yet different values for b. This way the lines are parallel. The lines don’t intersect, so there space no solutions and also the system is inconsistent. Due to the fact that the lines space not the exact same the equations space independent. The exactly answer is C. B) Incorrect. The 2 lines in this device have various slopes and different values for b. This way the lines intersect at one point. Because there is a solution, this system is consistent. And also because the lines are not the same, the equations are independent. The exactly answer is C. C) Correct. The two lines in this mechanism are the same; can it is in rewritten as . Because there are many solutions, this system is consistent. The present are the same so the equations room dependent. D) Incorrect. The two lines in this device have different slopes and also the exact same value for b. This method the lines crossing at one point—the yintercept. Recall that intersecting lines have actually one solution and therefore the mechanism is consistent. Because the lines room not the very same the equations room independent. The exactly answer is C. From the graph above, you can see that there is one systems to the mechanism y = x and also x + 2y = 6. The solution shows up to be (2, 2). However, you must verify an answer that you read from a graph come be certain that it’s no really (2.001, 2.001) or (1.9943, 1.9943). One method of verifying the the suggest does exist ~ above both currently is to instead of the x and also yvalues of the ordered pair right into the equation of every line. If the substitution outcomes in a true statement, climate you have actually the exactly solution!
Remember, that in stimulate to it is in a systems to the mechanism of equations, the value of the point must it is in a solution for both equations. As soon as you discover one equation because that which the point is false, girlfriend have figured out that that is no a systems for the system.
