Chapter 5: Systems

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Chapter 5 Questions and Comments misterlamb misterlamb 0 105 Dec 1, 2008 by misterlamb misterlamb

Summary Assignments:

5-1: Christian S. and Emma R.
5-2: Courtney G. and Brittani H.
5-3: Curtis K. and Nate J. (Bonus)
5-4: Courtney G. (Bonus) and Amanda T. (Bonus)
5-7: Amira A. (Bonus) and Andrew E. (Bonus)
5-8: Isaiah C. (Bonus) and Curtis K. (Bonus)
5-9: Ryan B. (Bonus) and Jessica R. (Bonus)
5-10: Jeremiah B. (Bonus) and Tyler Y. (Bonus)

Section 5-1: Inequalities and Compound Sentences

AA Section 5-1
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In Section 5-1: Inequalities and Compound Sentences, we first reviewed the symbols that are used in an inequality. These symbols are as follows:
>(greater than)
<(less than)
>_ (greater than or equal to (the lower line is actually directly under the greater than symbol))
<_ (less than or equal to (the lower line is actually directly under the less than symbol))
= / (not equal to (the slash is actually directly on the equal sign))
A compound sentence is a sentence where 2 clauses are connected by "and" or "or", and it applies in mathematics.
We also learned about intersections, which are the common elements of more than one set where the elements must be in both sets, and unions, which consist of elements in either one or both sets.
Intersections are represented by the symbol n, and they use the word "and."
Unions are represented by the symbol u, and they use the word "or."
We discovered how to show all of these things on a number line. For example:
x is greater than or equal to 5.
←————————→ x
The third section we learned was about the Properties of Inequality. It states that:
For all real numbers a, b, and c:
(Addition Property of Inequality) If a < b, then a + c < b + c.
and (Multiplication Properties of Inequality) If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
-Emma R.
Section 5-1: Inequalities and Compound Sentences we learned about Properties of Inequality,Inequalities,Compound Sentences,Intersection and Union.
Properties of Inequality says that 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
Inequalities are sentences that contain one of the symbols greater than or less than or greater than equal too or less than equal too and not equal too
Compound Sentences are sentences in which two clauses are connected by the word "and" or "or"
For example, 6<x and x<12 is a compound sentence
Intersection is a set consisting of values common to both sets(n)
Union is a set consisting of values in one or both sets(u)
-Christian S.

Section 5-2: Solving Systems Using Tables or Graphs

AA Notes 5-2
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In this lesson we learned how to solve systems using tables or graphs. A system is two sentences that are connected by "and". A solution set of a system is what is common in both of the equations, and normally it is just a point. We can solve systems with three different methods: graphing it by hand, making a table, or graphing it with a graphing calculator. You can use either method, but you should check it with a calculator because it's more accurate. You can also find the solutions to many real world situations.
Brittani H.

This lesson taught us how to use systems. Systems are sets of conditions that are combined by using the word "and". We learned that there are a few different ways to solve a system. Using tables, graphing by hand, and graphin by calculator are the ways taught in this lesson. Once the conditions have been combined, you have to use algebra to solve for the answer. After the answer is found it is always best to check the answer by plugging it in to the equation to make sure that you are correct.

- Courtney G.

5-3: Solving Systems by Substitution

AA Section 5-3
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Lesson 5-3 is yet another lesson on solving systems. This way is called substitution. Substitution is a fairly simple way of solving system. The first step in this process is to solve for one equation for one variable. Once you have the value of one variable, you can substitute that value in for the variable in the second equation. If you then solve that resulting equation, you find the value of the variable. You can then repeat the substitution process on the first equation using the value you found for the variable in the second equation. Again, this whole process is called substitution, and is used for finding the solution to a system.
Curt K.

This section involves solving systems by substitution. Their are techniques involved in solving these systems. One is using the Substitution Property of Equality that says if a=b then a may be substituted for b in any arithmetic or algebraic expression. If y is given, one could substitute it in an equation. Substitiution can also be used when solving 3x3 systems. We also talked about using substitution with a system that has one or more nonlinear equations. If one were to solve this they would have to write one equation for a single variable and substitute that expression into the other equation. Systems can also be consistent or inconsistent. If a system is inconsistent, it has no solutions. But if a system is consistent it has one or more solutions. -NJ

5-4: Solving Systems Using Linear Combinations

AA Section 5-4
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Amanda T.
In this lesson we learned how to solve a system using a linear combination. Earlier in the chapter we learned solving a system by table will just take a while and is like the guessing game, graphing by hand will only work with intergers, graphing by calculator is just too easy and is taking the easy way out and also by subsituting. In the lesson we learned two ways to solve a system by using linear combinations. The first way is by having a system will two variables and two equations. You simply eliminate a variable by combining the the equations. Next, solve for the variable thats left and then subsitute back to the original equation. After that solve for the other variable and once you find out what the other variable is you should check to make sure it's correct by plugging in both variables into the other equation.
We also learned how to solve a system by linear combination with three variables and three equations. You first group the equations so you have two sets of two equations. It doesn't which two equations you group together it can be the first and third and the second and first, ect.., it doesn't matter. Next, eliminate the same variable in each set. After that, set up a new system of two equations. By doing this you can use the next step like above that you used with two variables and two equations. Now you can plug the variables you found into the orginal equation that you used so you can find the third variable. Again, check your answer with all of the equations.

Courtney G.
This lesson showed us how to solve systems by using linear combination. When using this method, you have to add or subtract the given conditions and use algebra to find one of the given variables. once you find one of the variables, you plug the number into the equation in order to find the second variable. When you have a system that has three unknown variables and three different sets the first thing you have to do is group the first and second set and then group the second and third set together. Once you have done that you have to choose one (the same) variable in the two groups to eliminate by multiplying the sets by whatever number that when the sets added together will leave you with only two variables. Once you have two variables left in each group, you just follow the steps given for the "two by two" sets. After both variables are found, you plug those numbers into an original equation to find the third variable. Once you have found all three variables you should check your answer by plugging all three variables into the original equation to make sure the answers match.

5-7: Graphing Inequalities in the Coordinate Plane

AA Section 5-7
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In this lesson it was talking about graphing inequalities in the coordinate plane. To do this you have to draw a plane, and that line separates the plane on two distinct regions which are called half-planes. A half-plane is were either of the two sides of a line in a plane. You have to figurer out where to shade on the coordinate plane. The line is a boundary of both regions but it doesn't belong to either half plane. In inequalities and you have one variable can be graphed either on a number line or in the coordinate plane. Andrew E.

Amira A-
In lesson 5-7 we discussed graphing inequalities in the coordinate plane. We discussed half-planes, which are regions formed on a plane when a line is placed on it. We also discussed boundary lines, which are lines that divide the plane in half. There are three steps that we learned to graph linear inequalities. First- Treat inequalities as if they were equations, yet be mindful of the signs. Second- Graph the boundary line. (Dashed for >, <, and not equal to.) (Solid for less than or equal to, Greater than or equal to.) Third- Shade appropriately (check points) (when in “slope-intercept” form, the sign will tell you to shade above or below the boundary line.) We did a couple of examples then we learned what lattice points are, points whose coordinates are integers. Then we did some more examples, and received our homework which was page 316 #’s 1-26.

5-8: Systems of Linear Inequalities

Lesson 5-8 takes inequalities and puts them into systems. To graph these systems is similar to graphing regular systems. To begin, you treat the inequality signs as equal signs, but keep in mind you must either draw a solid or dotted line depending on whether the line is included in the feasible set or not. Once you have your lines drawn you must shade one of the two newly created half planes. The half plane that includes solutions to the equation is the one that must be shaded. You can determine this two different ways. The first way is to look at whether the inequality's solutions are less than or great than the graphed line. This is determined simply by the inequality in the equation. The second way to figure out how to shade is to just pick a point on a certain side and if it is a solution then whatever side that point is the side that is to be shaded. If you repeat this process for every equation in the system, you will eventually have a region where all the shading the all the equations overlaps. this is called the feasible set, and is the solution to the system of inequalities.
Curtis F. K.

5-9: Linear Programming I

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 one of the vertices of the feasible set. Profit Equation: the equation that allows us to determine the maximum.
You have to find 4 points that make up the feasible set. then shade the area that fits the appropriate equation.
-Jessica R.-Ryan B.

5-10: Linear Programming II

In this lesson we learned about Linear Programming II. There are 6 steps to solving a linear programming problems from scratch:1. Identify variables (if necessary)2. Organize information in a table3. Graph and find the vertices of the feasible set4. Write the profit equation5. Check the vertices to determine the minimum or maximum6. Interpret the results (write out your answer)We also learned how to graph the feasible set and find the vertices of a system by using INEQUALZ on your TI-84 calculator.
-Jeremiah B.

Linear programming on the calculator is a lot faster than doing it by hand. It also gives you a more accurate answer than doing it by hand. If you do it by hand you can not find the exact point and you have to set up an system to find the vertices. If you do it with the calculator if finds the points for you. Tyler Y.