Shortly we will investigate methods of finding such a solution if it exists. In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored.
The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y -intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system.
Thus, there are an infinite number of solutions. Another type of system of linear equations is an inconsistent system , which is one in which the equations represent two parallel lines. The lines have the same slope and different y- intercepts.
There are no points common to both lines; hence, there is no solution to the system. There are three types of systems of linear equations in two variables, and three types of solutions. We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines. There are multiple methods of solving systems of linear equations.
For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes. Graph both equations on the same set of axes, use function notation so you can check your solution more easily later.
You can check to make sure that this is the solution to the system by substituting the ordered pair into both equations. Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system. Plot the three different systems with an online graphing tool.
Categorize each solution as either consistent or inconsistent. If the system is consistent determine whether it is dependent or independent. You may find it easier to plot each system individually, then clear out your entries before you plot the next. Improve this page Learn More. Skip to main content. Module Systems of Equations and Inequalities. Search for:. Solutions of Systems Overview Learning Outcomes Identify the three types of solutions possible from a system of two linear equations.
It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. So I can see that m is the gradient of the lines and if the gradients are not equal that they will intersect.
But I am not sure if there is another way to demonstrate the above condition without referring to the geometrical explanation. Any help? The simplest way to show that a system has "a unique solution" is to actually find that solution! Sign up to join this community. The best answers are voted up and rise to the top.
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