2.10.2 The Graph of a System of Linear Equations
In a system of two equations, the graph of the linear system tells us important information about the algebraic solutions to the two equations:
|Different m||Cross once||One solution|
|Same m, different b||Never cross||No solution|
|Same m, same b||Same line||Infinite number of solutions|
Oftentimes, thinking about a mathematical creature algebraically can help us to understand it better. However, many times, by visualizing it on a graph we gain additional insights and a deeper understanding of the creature that we're studying. On the graph to the right is a "system of equations". We've already studied how to solve this system algebraically. What do you think our algebraic solution means graphically?
Take a look at the graph, play around with the algebra, and when you see the relationship between the graph and the algebraic solution of a system of equations, take a look here for a full discussion.
We can solve the system of equations by setting -3x+5=3x-1. Doing so, we get that -6x=-6 or that x=1. We can then plug 1 into either equation to get that y=2. So the solution to the system of equations is x=1, y=2 which can be thought of as the point (1,2).
Do you notice anything special about this point? It's exactly the point of intersection on the graph! Using an algebraic process, we've been able to determine something about the graph; once again algebra has shown itself to be a powerful tool in understanding how a mathematical creature behaves graphically.
The algebraic solution is the point of intersection on the graph.
Thinking about things from the other direction, we also see that if two lines (or actually two functions, in general), intersect on a graph, then the point of intersection will be the simultaneous solution to the system of equations. In other words, the picture tells us this without having to do any algebra!
With this in mind, use your imagination and sketch two lines on a graph. Can you visualize two lines that intersect once? Can you visualize two lines that don't intersect at all? If we can think about when these situations will occur, it will tell us when a system of linear equations will have no solution and when it will have 1 solution. Now we're using the power of the graph to help us to explain something algebraically!
Using the , find a second line will intersect, say, f(x)=2x+1 once and try to determine what has to be true about this second line, in general, in order for them to intersect.
In order for two lines to intersect once, the two lines need to have different .
Now let's try to figure out what has to be true in order for the two lines to never intersect; if we can determine when this happens graphically, we'll be able to tell when a system of linear equations will have no solutions!
Using the , determine when a second line will intersect, say, f(x)=-2x+3 never.
Two lines that don't intersect have the slopes and y-intercepts.
Based on what we've discovered, we now see that a system of linear equations can either have no solution or one solution. Now use your imagination again and see if you can visualize two lines that cross twice, which would give us a system of equations with two solutions.
Two lines can't cross twice, three times, or four times....So, are we stuck with either 0 or 1 solution?
Can you find a second line which crosses the first line 2y=2x+4 an infinite number of times?
(Please write in your answer in the form y=mx+b)
y=x+2 will do the trick for us. In other words, we can just rewrite the first line in the form y=mx+b by dividing both sides by 2. This may seem a bit like "cheating", but it illustrates the point that the only way for a system of linear equations to have an infinite number of solutions is for them to have the same exact slope and y-intercept.
|Situation||Graphical Meaning||Algebraic Meaning|
|Different m||Lines cross once||System of equations has one solution|
|Same m, different b||Lines never cross||System of equations has no solution|
|Same m, same b||It's the same line!||Infinite number of solutions|
We've just taken advantage of the powerful relationship between graphs and equations. Think about how difficult it would have been to come up with the table above if you had to think about things purely from an algebraic perspective!
Now that we have had some exposure to working with two lines, let's play around a little bit. In the next section, we'll study parallel and perpendicular lines, which are two lines which relate in a particular way.