Section 9.1: Systems of Equasions

Today, we reviewed systems of equasions. Remember that there are three ways to solve a system of equasions:

Substitution: solve one equasion for a variable, and plug it in to the other. For example:

2x + y = 13                                     x - y = 5

y = 13-2x   Solve for one variable.               x - (13 - 2x) = 5    substitute 13-2x for y.


Once we have this, we can solve for x.

3x - 13 = 5    Simplify.


3x = 18    isolate x.

x = 6

Once we have determined x, we plug x back into the first equasion.

y = 13 - 2(6)    plug in x.

y = 13 - 12     simplify.

y = 1

Now that we know the x and y values, we can form an ordered pair which is the solution to the equasion, or where the two graphs intercept. Our ordered pair is (6,1), which is the solution to the equasion.


Now for the next method.

Elimination: multiplying one equasion by a certain number so that when you add the two equasions together, one or more variables cancel out. For example:

2x + y = 13                                   x + 2y = 5

Now we have to multiply one of the two equasions by a certain number so one or more variables cancel out. Lets multiply the first equasion by -2. Remember to multiply both sides of the equasion. We end up with:

-2(2x + y) = -2(13)

-4x - 2y = -26

Now we can add the equasions, and something will cancel out. In this case, the y's will cancel out.

    -4x - 2y = -26
+    x + 2y = 5
_______________
           -3x = -21

Notice that the y's canceled out. Now we can solve for x, and later, y.

-3x = -21

x = 7                              x + 2y = 5

                                      2y = 5 - 7
                                      
                                      2y = -2
                                     
                                      y = -1

Now our solution has an x-value of  7 and a y-value of -1, which forms the ordered pair (7,-1).


The third method is graphing. for this, enter both equasions. lets use 2x + y = 13 and x - y = 5. in order for it to work on our calculator, we need to isolate y. We end up with:

y = -2x + 13                         y = x - 5

Now, for all of you TI users, graph the functions and under the calc menu find the intercept.

We find that the x-value is six, and the y-value is one, which gives us the ordered pair (6,1) as the solution.

It is also possible to solve for more than two equasions, it just involves an extra step in substitution or elimination.


Some helpful links that Mr. Wilhelm should show the class to enrich our learning:

http://www.youtube.com/watch?v=LhXkYfLgRoA skip to 1:06

http://www.youtube.com/watch?v=2qdql9vsWWM

Thats all for now. Good luck on Friday for all of those who need it for the AP test!
-Jacob