Algebra Period 4

Inequalities in Two Variables 9-5


You will shade an x y graph to find the side of a linear equation that fits the solution

1. Graph the inequality by graphing the line with an x y table or y = mx + b

2. For inequalities that do not have the equal sign (no crayon!), make a dotted line; for inequalities with an equal sign, make a solid line

3. Shade the side of the line that works in the inequality (is a solution)

ONE SIDE WILL WORK, THE OTHER WILL NOT!
I always check the point (0, 0) first if possible because it's the easiest point to check :) If (0, 0) works, shade that side. If (0, 0) doesn't work, shade the other side.

4. Check the other side just to make sure that it does not work.


EXAMPLE: x + y > 5

You graph the line with DOTTED line because it cannot be equal to 5.
I would put it in y = mx + b format: y > -x + 5 
(Remember that if you multiply or divide by a NEGATIVE to isolate the y, you'll need to switch the symbol!)

You pick an easy point on one side of the line and substitute to see if that side is a solution.

If that does not work, pick an easy point on the other side to see if that side checks.




Systems of Inequalities in Two Variables 9-6
If there is more than 1 Inequality (a system of inequalities),

1. Follow the same procedure as above for one equation
 but you will need to do it for each equation.

2. Use a different type of shading for each so you won't get confused

(Ex: Use slanted lines one way and then slanted lines the other way. Use different colors if possible. Make one set of lines wavy and the other set straight)


3. Where the 2 shadings overlap each other is called the solution of the system of inequalities.
(any point in the overlap should work in BOTH inequalities - make sure you check all the inequalities in the system!!!!!)


 Solving Systems of Equations 8-1 to 8-3
(2 equations with 2 variables)

You cannot solve an equation with 2 variables - you can find multiple coordinates that work

TO SOLVE MEANS THE ONE COORDINATE THAT WORKS FOR BOTH EQUATIONS

There are 3 ways to find that point:

1. Graph both equations: Where the 2 lines intersect is the solution

2. Substitution method: Solve one of the equations for either x or y and plug in to the other equation

3. Addition method: Eliminate one of the variables by multiplying the equations by that magical number that will make one of the variables the ADDITIVE INVERSE of the other


Example solved all 3 ways:

Find the solution to the following system: 
2x + 3y = 8 and 5x + 2y = -2


1. GRAPH BOTH LINES: Put both in y = mx + b form and graph

Read the intersection point....You should get (-2, 4)


2. SUBSTITUTION: Isolate whatever variable seems easiest

I will isolate y in the second equation: 2y = -5x - 2

y = -5/2 x - 1 
Plug this -5/2 x - 1 where y is in the other equation
2x + 3y = 8
2x + 3(-5/2 x - 1) = 8

2x - 15/2 x - 3 = 8

4/2 x - 15/2 x - 3 = 8

-11/2 x - 3 = 8

-11/2 x = 11
-2/11(-11/2 x) = 11(-2/11)
x = -2

Plug into whichever equation is easiest to find y



3. ADDITION: Multiply each equation so that one variable will "drop out" (additive inverse) 
I will eliminate the x, but I could eliminate the y if I wanted to!
 
2x + 3y = 8 and 5x + 2y = -2
 
Notice that there is 2x in first and 5x in 2nd equation.
To make the x terms additive inverses, I'll multiply both sides of each equation so that the x terms are both 10x. 
I'll need to make one +10x and the other -10x:
5(2x + 3y) = (8)5

-2(5x + 2y) = (-2)(-2)

Now when I distribute I get:
10x + 15y = 40

-10x - 4y = 4


ADD TO ELIMINATE THE x term:

11y = 44
SOLVE: y = 4

Plug into whichever equation is easiest to find x

NOTICE THAT FOR ALL 3 METHODS, THE SOLUTION IS THE SAME!
THEREFORE, USE WHATEVER METHOD SEEMS EASIEST!!
 
2 special cases!
NO SOLUTION: When would 2 lines never intersect???

When they're PARALLEL! So always check first to see if the lines have the SAME SLOPES. If they do, you're wasting your time trying to solve the system because the solution if the point where they intersect and there is none!
 
INFINITE SOLUTIONS: When would 2 lines have infinite points in common???
When they're multiples of each other and therefore are really THE SAME LINE! If you make each line y = mx + b first, you'll find this out quickly. Then just say INFINITE SOLUTIONS.