Friday, December 14, 2007

Graphing Systems of Equations

The goal of solving systems of equations is to determine if two or more lines intersect and if they do where do they intersect. This lesson will only deal with two lines.

First we need to determine how the lines of the two equations relate to each other. Do they intersect, are they parallel, or are they the same line. The mathematical ways of describing this is are they consistent, inconsistent, dependent, and independent. There are three options as shown in the picture below: To find the answer to the system you need to ask one or two questions depending on the problem.


Intersecting lines: Independent and consistent


Parallel lines: Independent and inconsistent


Same line: Inconsistent

Question 1: First write the equation in slope intercept form. Then ask the question is the slope in each equation the same or different? If it is the same then they are parallel and you need to proceed to the next question. If it is different then the point at which the lines cross is the solution to the system and we are done.

Question 2: Since the slope is the same the lines are parallel. Now you need to ask the question "Are they the same line?" If the y-intercepts are the same then they are the same line and there is an infinite number of solutions. The line is the solution to the system of equations because it represents all of the points that work in both equations which are really the same equation just written differently.

If they are not the same then there is no solution, that is the lines do not cross.



Practice Problems:

Find the solution of the equations y = 2x + 4 and y = 0.5x - 2

First we recognize that the equations are in slope intercept form (y = mx + b) and that the slopes of each of the lines are different so we know that they will cross at some point. The next step is to graph the line. It is very important that your lines be accurate so I would recommend placing as many calculated points as possible on the graph for each line.

The solution is ( -4, -4 ). Make sure to check your answer by plugging the point into both problems.
y = 2x + 4
-4 = 2 ( -4) + 4
-4 = -8 + 4
-4 = -4

and

y = 0.5x - 2
-4 = 0.5 (-4) - 2
-4 = -2 - 2
-4 = 4

Lets try another example:
Solve the system of equations:
y= 4
2x - 3y = -6

First off notice that the second equation is in standard form and not slope intercept form. We need to rearrange it so that we can compare the slopes.
2x - 3y = -6
Subract 2x from both sides.
-3y = -2x - 6
Now divide both sides by -3
y = (2/3)x + 2

Again notice that the slopes are not the same so we know that they will cross at some point. Graph your line, making sure to plot lots of points to keep it accurate.

The solution is ( 3, 4). Check your answer
y = 4
4 = 4. Remember that this equation doesn't care about the x value so it can be anything.

2x - 3y = -6. Always use the original problem in case you made a mistake when you were changing the problem to slope-intercept form.
2(3) - 3(4) = -6
6 - 12 = -6
-6 = -6

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