Graphing Systems of Linear Equations Learning Objective(s) · Solve a system of linear equations by graphing. · Determine whether a system of linear equations is consistent or inconsistent. · Determine whether a system of linear equations is dependent or independent. · Determine whether an ordered pair is a solution of a system of equations. · Solve application problems by graphing a system of equations. Recall that a linear equation graphs as a line, which indicates that all of the points on the line are solutions to that linear equation. There are an infinite number of solutions. If you have a system of linear equations, the solution for the system is the value that makes all of the equations true. For two variables and two equations, this is the point where the two graphs intersect. The coordinates of this point will be the solution for the two variables in the two equations. The solution for a system of equations is the value or values that are true for all equations in the system. The graphs of equations within a system can tell you how many solutions exist for that system. Look at the images below. Each shows two lines that make up a system of equations.
When the lines intersect, the point of intersection is the only point that the two graphs have in common. So the coordinates of that point are the solution for the two variables used in the equations. When the lines are parallel, there are no solutions, and sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions. Some special terms are sometimes used to describe these kinds of systems. The following terms refer to how many solutions the system has. o When a system has one solution (the graphs of the equations intersect once), the system is a consistent system of linear equations and the equations are independent. o When a system has no solution (the graphs of the equations don’t intersect at all), the system is an inconsistent system of linear equations and the equations are independent. o If the lines are the same (the graphs intersect at all points), the system is a consistent system of linear equations and the equations are dependent. That is, any solution of one equation must also be a solution of the other, so the equations depend on each other. The following terms refer to whether the system has any solutions at all. o The system is a consistent system of linear equations when it has solutions. o The system is an inconsistent system of linear equations when it has no solutions.
From the graph above, you can see that there is one solution to the system y = x and x + 2y = 6. The solution appears to be (2, 2). However, you must verify an answer that you read from a graph to be sure that it’s not really (2.001, 2.001) or (1.9943, 1.9943). One way of verifying that the point does exist on both lines is to substitute the x- and y-values of the ordered pair into the equation of each line. If the substitution results in a true statement, then you have the correct solution!
Remember, that in order to be a solution to the system of equations, the value of the point must be a solution for both equations. Once you find one equation for which the point is false, you have determined that it is not a solution for the system.
Graphing as a Solution Method You can solve a system graphically. However, it is important to remember that you must check the solution, as it might not be accurate.
Graphing a Real-World Context Graphing a system of equations for a real-world context can be valuable in visualizing the problem. Let’s look at a couple of examples.
Note that if the estimate had been incorrect, a new estimate could have been made. Regraphing to zoom in on the area where the lines cross would help make a better estimate.
A system of linear equations is two or more linear equations that have the same variables. You can graph the equations as a system to find out whether the system has no solutions (represented by parallel lines), one solution (represented by intersecting lines), or an infinite number of solutions (represented by two superimposed lines). While graphing systems of equations is a useful technique, relying on graphs to identify a specific point of intersection is not always an accurate way to find a precise solution for a system of equations. |