Bonus Assignment

Hey Bob, you missed out today on our 5.1 lesson. Luckily, since I'm such a great friend, I'm willing to help you out.

We learned how to solve systems of equations by graphing. Now that may sound difficult, but it's actually quite simple. First and foremost, what is a system of equations? A system of equations is basically a set of equations that use the same variables. So for example, take these two equations (keep in mind they are in the same system):

1) y=3x+2
2) y=9x+10

Since these two are part of the same system of equations, the X and Y for both equations are the same.
The next question you may have is: How in the world would you solve something that has two different variables?
Now normally, you wouldn't be able to. However, these two equations are part of the same system (and I can't stress that enough), so, you can. There are many different ways of doing it, but the method we learned in class was through graphing. Basically, what you do is graph the two lines, and wherever the two lines meet, that's our solution! See, it's very easy! Oh and in case you forget how to graph a line, here's two simple ways:

1.Remember back in Gr.9 and Gr.10 math where we learned the equation of a line: y=mx+b. Well you just use that. In the equation, m is the slope (a negative m indicates the line is going down), and b is the y-intercept (where the line hits the y axis...the vertical one.) You start with the y intercept, and start plotting using the slope. Now remember, slope is equal to rise/run. So if the slope is 5, then you would move one square over, and 5 square up.

2.The second way is a little less complicated, but in my opinion, is very time consuming. You pick a value for X (x=1, x=2, x=3, etc) and substitute it into the equation. Then you solve for Y. Once you have the two coordinates, plot the point on the graph. Pick a few more points, and once you have enough, join the points with a line.

Now, the coordinates of the point where the two lines intersect is our answer. For the two equations above, they intersect at: (-4/3 , -2) So that is our answer for this system of equations: X= -4/3 and Y= -2
Easy enough right?
Now that you understand how to solve it, you may wonder how it works; why is the point where the two lines meet the answer? That point, (-4/3 , -2), where the two lines intersect, is on both lines. If you substitue X= -4/3 and Y= -2 back into both equations, they will have the same answer. That point is the point where the two lines are "equal," and thus they intersect (This'll be proven next chapter when we solve algebraically).

Unfortunately, not all the systems of equations we graphed were linear (just a line). Some of them were parabolas too. These were referred to as non linear systems. Remember though, a non linear system is any graph that isn't a line. So, a cubic function would be a non linear system as well.

Take for example:
(also on pg 305)

1) y=x^2
2) y=x + 2

After solving this graphically, you'll see that there's two solutions (-1, 1) and (2, 4).

Why is it like this? Well as you can see on the graph, the linear equation (equation 2) intersects the parabola (equation 1) in two places, at (-1, 1) and (2, 4). Hence, there are two answers.

Well, that's pretty much what we learned today in class. Hope this helps you. Next day we're going to learn how to solve this without a graph. Although using a graph is easier, and there's a smaller chance of messing up, sometimes the lines will intersect at like -50, 30 or something like that. Also, graphing is time consuming.

A Colourful Introduction to the Wonderful World of Complex Numbers.

1. The real number system has been adequate in dealing with almost everything - but what about numbers with negative roots? (Un)Luckily, complex numbers were created. The concept of complex numbers first surfaced around the 16th century, in the midst of the Renaissance. The idea of of an "imaginary number" was a preposterous notion - in fact, the term "imaginary" was coined as a sign of disrespect. Nevertheless, the idea was gradually accepted by mathematicians around the world (headed by headed by a description of the geometric sequence created by Caspar Wessel). Complex numbers are a system of numbers that has been developed to deal with negative roots. A complex number (in the form a + bi), is actually an extension created by assigning a unit, i, which satisfies that i^2 = -1. In this form, the a and b are real numbers. Using complex numbers, we are now able to solve negative roots.


2. As ridiculous as it may sound, complex numbers actually have many uses in the real world. In some of the sciences, especially physics, complex numbers sneakily creep their way in. There are only a few cases where complex numbers are directly used. For example, in electrical circuits, imaginary numbers can be used to easily determine the changes in voltage and current.Also in electromagnetism, complex numbers are used. In most cases however, complex numbers aren't directly used, but instead, the concept of complex numbers is involved. Some examples of these can be found in engineering and architecture, where the concept of complex numbers is extremely useful.


3.1) (3+2i)^2 - 6 (3+2i) + 13 = 0
9 + 12i + 4i^2 - 18 - 12i + 13 = 0
4 + 4i^2 = 0 (i^2 = -1)
4 - 4 = 0

(3-2i)^2 - 6 (3-2i) + 13 = 0
9 - 12i + 4i^2 - 18 + 12i + 13 = 0
4 + 4i^2 = 0 (i^2 = -1)
4 - 4 = 0

2a) a = 1 b = 4 c = 5
-4 ± ((4)^2 - 4(1)(5))^1/2 / 2(1)
-4 ± (-4)^1/2 / 2
-4 ± ((4) (-1))^1/2 / 2
-4 ± ((4)^1/2 (-1)^1/2 )/ 2
-4 ± 2i / 2
-2 ± i

2b) a = 1 b =-2 c = 3
2 ± ((-2)^2 - 4(1)(3))^1/2 / 2(1)
2 ± (-16)^1/2 / 2
2 ± ((16) (-1))^1/2 / 2
2 ± ((16)^1/2 (-1)^1/2 )/ 2
2 ± 4 i / 2
± 2i

3) x^2 + 2x + 2
a = 1 b = 2 c = 2
-2 ± ((2)^2 - 4(1)(2))^1/2 / 2(1)
-2 ± (-4)^1/2 / 2
-2 ± ((4) (-1))^1/2 / 2
-2 ± ((4)^1/2 (-1)^1/2 )/ 2
-2 ± 2 i / 2
-1 ± i

Check:

(-1 + i)^2 + 2 (-1+i) + 2 = 0
1 - 2i + i^2 - 2 + 2i + 2 = 0
1 + i^2 = 0
1 - 1 = 0

(-1 - i)^2 + 2 (-1 - i) + 2 = 0
1 + 2i + i^2 - 2 - 2i + 2 = 0
1 + i^2 = 0
1 - 1 = 0

Sources:

http://mathforum.org/library/drmath/view/53844.htm
http://www.math.toronto.edu/mathnet/questionCorner/complexinlife.html
http://en.wikipedia.org/wiki/Complex_numbers
http://www.purplemath.com/modules/complex.htm