Chapter 5: Systems


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Chapter 5 Questions and Comments misterlamb misterlamb 1 136 Dec 8, 2008 by Dakota13 Dakota13

Summary Assignments:

5-1: Dakota E. and Shannon M.
5-2: Ricky S. and Gage A.
5-3: Brittany K. and Marissa B.
5-4: Sarah P. and Kyle L.
5-7: Tarah L. and Autumn T.
5-8: Mitch K. (Bonus) and Dale N. (Bonus)
5-9: Austin L. (Bonus) and Dakota E. (Bonus)
5-10: Autumn T. (Bonus) and Nick H. (Bonus)

Section 5-1: Inequalities and Compound Sentences


AA Section 5-1
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In 5-1 Students Learned about,
Intersection & Union
Inequality & Compound Sentence
New Properties and Tips

Intersection- Common elements of more than one set; Elements must be in both sets (AND)
Notation: is an upside down U <--- so turn that upside down, uhm or flip it I don't know...
Union- Set consists of elements in either one or both sets (OR) Notation: U (hey what do you know A U for UNION, isn't that funny?)

Inequalities are sentences that contain symbols (Less[Greater] than [or equal to], and not equal to)
Compound Sentences are sentences where two Clauses (Pay attention in english) are connected by "and" or "or"

PROPERTIES OF INEQUALITY
-For all real numbers a, b, and c

Addition Property of Inequality
-If a is less than b, then a+c is less than b+c
Proxy-Connection: keep-alive Cache-Control: max-age=0 ltiplication Properties ofProxy-Connection: keep-alive Cache-Control: max-age=0 0Inequality
-If a is less than b, and c is greater than 0, then ac is less than bc
-If a is less than b, and c is less than 0, then ac is less than bc

*When multiplying or dividing by a negative flip inequality sign*
like Less than to greater then, so on...

-Dakota E.

Intersection and Compound Sentences
Properties of Inequality
For all real numbers a,b, and c...
If a<b, c>0, then a+c<b+c (Addition Property of Inequality)
If a<b, c>0 , then ac<bc (Multiplication Property of Inequality)
If a<b, c<0, then ac>bc
Inequality
An inequality is a sentence that contains one of the symbols <,>, greater than (or less than) or equal too, and not equal too
Compund Sentences
A Compound sentence is a sentence in which two clauses are connected by the word "and" or by the word "or"
For example, 4<x and x<8 would be a compound sentence
Intersection and Union
Intersection set is a set consisting of values common to both sets and Union sets are sets the consist of values in one or both sets
by Shannon M.

5-2: Solving Systems Using Tables or Graphs


AA Notes 5-2
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Section 5-2 Summary
Richard S.

System: two sentences connected by “and”
Notation: {y=4x+8}
Solution Set of a system: since we’re using “and”, we are looking for what
*often it`s just a point*
So now how can we solve a system?
1. You could try a table, which, is the last way you should try and get it because it is very time consuming and it`s pretty much guess and check.
2. You could try graphing, which, will show you a general area of where the lines will intersect however it is not very accurate.
3. The best way is to use your graphing calculator, it will provide you with the accurate intersection point and is quite easy to do.
STEPS FOR A GRAPHING CALCULATOR
1. Press [Y=]
2. Next put in your equations
3. Press [graph] [2nd] then [trace]
4. Choose option 5: intersect
5. Put the cursor on a line and press [enter] on it do the same for the 2nd line
6. Move your cursor close to where you think the intersect is then hit [enter]
Hope this helped you to understand but that sums up section 5-2

5-3: Solving Systems by Substitution



AA Section 5-3
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sometimes tables and graphs enable you to find solutions and to find the exact solution you need to use algebraic techniques.
Substitution Property of Equality- states that if a=b, then a may be substituted for b in any arithmetic or algebraic expression.
inconsistent systems- has one or more solutions.
consistent systems- has no solutions at all.
substitution may be appropriate when one of the following applies.
1. at least one of the equations has beed or can easily be solved for one of the variables.
2. there are two or more linear equations and two or more variables
3. the system has one or more nonlinear equations.

-marissa b &&& BRITTANY K.

5-4: Solving Systems Using Linear Combinations


AA Section 5-4
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Sarah P.
Recap: How can we solve a system?1. Tables 3. Graphing calculator2. Graphing(by hand) 4. Substitution

Process to solve 2x2's by combination:
1. Eliminate a variable(your choice) by combining equations.
2. Solve for that variable.
3. Substitute back into one of the original equations.
4. Solve for the other variable.
5. Check in the other equation.

Process to solve 3x3's by combination:
1. Group equations so you have two sets of two equations.
2. Eliminate the same variable in each set.
3. Set up a new system of two equations.
4. Solve the 2x2 system.
5. Plug back into an original equation to the third variable.
6. Check your answer in all equations.

5-7: Graphing Inequalities in the Coordinate Plane


AA Section 5-7
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In this lesson we learned about graphing inequalities on coordinate plane. The line separating the into two distinct regions called half planes. The line its self is the boundary of the two regions. The Inequalities can be on a number line or a coordinate plane.
On a number line
1. All the points on a number line according to the inequality
2. Make sure the circle closes or is open according to the inequality

On a coordinate plane:
1. Plot all point according to the in equality
2. Shade the appropriate region on the plane

Autumn T.:)
=Tarah L.
The region that is created on a plane when you input a line is called the Half-Plane.The line that determines both halves of the plane is called the Boundary Line.~The key for this lesson would be to pay attention to the sign of the inequality. The inequality tells you whether your boundary line should be solid or dashed & where to shade, whether it be above or below the line.-If the inequality is "less than" (<), "greater than" (>), or "not equal to", then the boundary line should be dashed.-If the inequality is "less than or equal to" or "greater than or equal to", then the boundary line should be solid.As for the shading, like what was said earlier, it all depends on the inequality sign.For example, say that:y>5If you were to graph this, you would shade above your boundary line. Why? When you shade above the boundary line, it shows all of the values that makes this equation true, in other words, all the numbers that are more than 5 are all shaded, which makes it true.

5-8: Systems of Linear Inequalities


MitchK-Lesson 5-8 combines systems and linear inequalities(and graphing them). We learned how to graph inequalities on planes. Now we just graph several inequalities on a plane. To graph systems of inequalities, just graph all the inequalities in the system. Remember to treat them as if they're equations, but shade above/or below the lines. Do this for each one until you find the region where all inequalities of the system overlap. This region is called the feasible set or feasible region of that system. Once you've found the feasible set, there will be boundaries and vertices. The boundaries can be thought of as the walls that outline the feasible region. The feasible region doesn't always have to be a closed shape; it may have two boundaries extending infinitively in two directions, therefore have one vertice. You will be asked to find the coordinates of the vertice(s) and to do this just apply what we learned earlier in the chapter about solving systems of equations to systems of inequalities. Actually it's the same. Pick the lines you want to find the verticy of. Then switch the inequality signs of those lines to equal signs and solve using one of the methods.

Dale N.- Vocab
Feasible Set - set of solutions to a system of linear inequalities
Feasible Region - " "
Vertices - intersection of the boundaries

In 5-8, we learned how to graph multiple inequalities in a system. It was a step ahead of 5-7 which we learned how to graph single inequalities. Once you graph your inequalities, you will have a common shaded point. Your common region, or your feasible region, is your answer. Also, the points around your feasible region are you vertices. Example : y > 1.5x - 7.5, y < 5 - x/8, x > 2. You graph your inequalities like you graph equalities.(5-7).
Once you graph and do your shading you find your feasible region. Once you have found that you can find your vertices. After you have found your feasible region and your vertices you are done.
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example
Graphing inequalities example

(www.dplot.com/ systemsofinequalities.htm)

5-9: Linear Programming I


5-9 was about linear Programming as you can see ^^^
its not much different than previous lessons in the chapter. Looking at the feasible sets we graph the points, then look at the Verticesagain looking at the inequalities shade accordingly.
Dakota E.

Like Dakota said 5-9 is not much different from the previous lesson. This lesson is about linear programming and there is some vocab that you need to know first.
Linear Programming Problems:problems that lead to a system of inequalities that allow us to find a maximum or minimum.
Linear Programming Theorem:the feasible set of every linear programming problem is convex and the maximum or minimum quantity is determined at 1 of the verticies of the feasible set.
Profit Equation:the equation that allows us to determine the maximum or minimum quantity.
In this lesson you will often use this to find out what combination of items will give the most profit or the least profit. To find the coordinates you will use points on the feasible set.
Austin L.

5-10: Linear Programming II

In this lesson we learned how ho solce linear programming problems from scratch. In order to solve a linear-programming problem you would follow these steps.
1. Identify the varibles.
2. Identify the constraints, and translate them into a system of inequalities relating the variables. If necessary, make a table
3. Graph the system ot inequalities; find the vertices of the feasible set.
4. Write a formula or an expression to be maximized or minimized.
5. Apply the linear-programming theorm,
6. Interpret the results.
by: Autumn T. :)



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