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Grade 8 Extension Menu

Writing Linear Systems and Systems of Inequalities

Choose a learning activity from one square to complete. If you choose the square, “Write your idea here,” please see the teacher with your idea first.

Circle the number of the learning activity you choose.

Turn in this paper with your work.

|1. When a system with three or more linear |2. Write three different systems that each have | 3. Design a brochure or foldable to explain |

|inequalities is graphed, polygons can be formed. |a solution of x=8 and y=22. One system should be |three different methods that can be used to solve|

| |appropriate for solving using graphing, one for |a system of equations. Include details about |

|Think about and describe what regions formed by |solving using substitution, and the last for |what types of systems or situations are best |

|the intersection of nonlinear inequalities would |solving using combinations (elimination). |solved using those methods. |

|look like. | | |

| |Optional Challenge: For each system that you | |

| |write, design a real life situation that could be| |

| |modeled by that system. | |

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|4. Use the “Solving Systems with Matrices” |5. Complete the 7-5 Enrichment worksheet by |6. |

|worksheet to learn how to solve a system of |writing systems of inequalities to represent each|Write your idea here |

|linear equations using matrices. If you are not |shaded region. | |

|familiar with entering and using matrices on the |On a coordinate plane, create your own shaded | |

|graphing calculator, visit the website |region and write the system of inequalities | |

| by it. | |

|s/matrices.htm | | |

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Teacher Resource Page

Grade 8

Extension Menu

Concept and/or Topic: Linear Systems and Systems of Inequalities

Intended Purpose: Culminating activity for the unit or alternative activity for students who have mastered curricular indicators.

Standards:

CCSS.Math.Content.8 Analyze and solve linear equations and pairs of simultaneous linear equations.

CCSS.Math.Content.8.EE.C.7 Solve linear equations in one variable.

CCSS.Math.Content.8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

CCSS.Math.Content.8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

CCSS.Math.Content.8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Organizational Tips:

Box 3: Information about foldables can be found in your Glencoe resource kit.

Box 4: Duplicate the Solving Systems with Matrices handout and make available for students.

Box 5: Duplicate the 7-5 Enrichment handout from your Glencoe Resource kit and make available for students.

Solving Systems with Matrices

The beginning of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However, it was not until near the end of the 17th century that the ideas reappeared and development really got underway.

It is not surprising that the beginning of matrices should arise through the study of systems of linear equations. The Babylonians studied problems which led to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:-

There are two fields whose total area is 1800 square yards. One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field.

The Chinese, between 200 BC and 300 BC, came much closer to matrices than the Babylonians. Indeed, it is fair to say that the text Nine Chapters of the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods.

To solve a system of linear equations using matrices, you can multiply the inverse of the coefficient matrix by the constant matrix.

____________________________________________________________________________________System of Linear Equations

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Coefficient Matrix

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Constant Matrix

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Solution to the system:

-1

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