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Lesson 133: Graphing Systems of Linear Equations
In this very important lesson we'll learn more about graphing systems of linear equations. All we have to do is graph two different lines on the same coordinate plane, and see where they intersect. It is easy once you know what to do.
Before starting this lesson, make sure that you review all of the previous lessons on graphing linear equations. Make sure that you fully understand them, or you will have difficulty with this lesson.
Very often we are asked to graph a system of linear equations. Sometimes this is referred to as solving a system of linear equations graphically. We're given two linear equations, just like the kind we've been working with. What we're asked to do is graph both of them on the same coordinate plane, and determine the point at which they intersect. The coordinates of that point satisfies both equations simultaneously. The x and y values will work in both equations. Sometimes we refer to these problems as a system of simultaneous equations. We want a value of x and y that work in both equations at the same time.
On
the graph at left, I've graphed two different linear equations like
we've learned to do. In blue we have y = 3x - 2, and in red we
have y = -x + 2. Review the previous lessons on graphing
linear equations if you're not sure how I graphed these lines.
Now that they are graphed, I can see that they intersect at (1, 1). That means that x =1, y = 1 will work in both equations. If you substitute those numbers into each equation, you will see that they work. In fact, no other set of coordinates work, as you can see from the graph.
Note that if you are using graph paper and pencil to do your graphing, you'll need to be very careful. Plot several points along each line, and then connect them using a sharp pencil and a straight edge, like a ruler. Once you see where they intersect, always check the point in both original equations to see if it works in both.
There are some special cases that we need to be aware of. First, if one equation is a multiple of another, then the two lines are actually the same line. For example, let's say one line is y = 2x + 3, and the other is 3y = 6x + 9. If we were to multiply both sides of the first equation by 3, which we're allowed to do, we would get the second equation. Or, if we divided each side of the second equation by 3, we would get the first one. That means that the two equations are actually the same thing. The solution to the system is just the equation itself, where x can be any real number.
Another special case is where we have two lines that have the same slope, but different y-intercepts. Try to picture that on a graph. If two lines have the same slope, they are parallel. They have the same steepness, so they will never intersect. There is no value of x and y which will satisfy both equations at the same time. We can say that the solution to such a system of equations is the empty set, often written as { }.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.