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Systems of Equations An Opening, Work Session, and Closing Using the TI-84 and Navigator The Opening LearningCheck Warm-Up Results • Are there any misconceptions? Georgia Performance Standards • M8A5. Students will understand systems of linear equations and inequalities and use them to solve problems. – a. Given a problem context, write an appropriate system of linear equations or inequalities. – b. Solve systems of equations graphically and algebraically, using technology as appropriate. – c. Graph the solution set of a system of linear inequalities in two variables. – d. Interpret solutions in problem contexts. Essential Questions • How can I interpret the meaning of a “system of equations” algebraically and geometrically? • What does it mean to solve a system of linear equations? • How can the solution to a system be interpreted geometrically? • Why is graphing a system of inequalities a good way to show the solution set? • How can I translate a problem situation into a system of equations or inequalities? • What does the solution to a system tell me about the answer to a problem situation? The Work Session Creating Triangles, Using TI-84 and the Navigator Activity: “Try”-angles • If your calculator number is odd, move to the left side of the room. You will be the RED group. • If your calculator number is even, move to the right side of the room. You will be the BLUE group. “Try”-angles • RED Group – Using the red paper, straight edge, and compass in front of you, create right triangles where one of the angles, x is twice as large as the other angle, y. Each member must have unique angle measures for x and y. – Cut out your triangles and place them on the grid paper in the appropriate place. • Blue Group – Using the blue paper, straight edge, and compass in front of you, create triangles where two of the angles, x and y, are complementary. Each member must have unique angle measures for x and y. – Cut out your triangles and place them on grid paper in the appropriate place. “Try”-angles • Using your keyboards and complete sentences, make some observations using mathematical lingo about what you see on the grid paper. Try-angles • Using your TI-84, logon to NavNet. • Plot your triangle’s “point”. • Input an equation that would accurately represent the points your group plotted. Try-angles • Going back to your keyboards, answer the following: – 1) What do you notice about the equation, the points, and the initial directions given to your group? – 2) Are there any other triangles that could have been “plotted” that are not already on the grid paper? How do you know? – 3) Create a triangle that could be placed in the fourth quadrant. What do you notice? – 4) Create a triangle that could be plotted in the second quadrant. What do you notice? – 5) What is significant about the the red and blue triangles that overlap? System of Equations • Each of the initial instructions for the red and blue group can be translated into equations: – x = 2y (Left or Red Group) – x + y = 90 (Right or Blue Group) • You noticed after “plotting” the two points that one of the red triangles and one of the blue triangles overlapped. • This overlapping is consequently where the two triangles have degree measures of 30° and 60°, called the solution. • Because both equations are true for x = 30° and y = 60° at the same time, they are considered to be a system of equations. Substitution Method • Another method for solving a system of equations is by substituting one variable in one of the equations for the variable expression in the other. – x = 2y – x + y = 90 • Because the first equation is solved for x, you can substitute the expression 2y in the second equation for x. • (2y) + y = 90 • Notice now that the expression on the left can be simplified and y can easily be solved for. • 3y = 90 • y = 30 • Now that y has been determined, to solve for x, substitute the value for y into either equation. Substitution Method • X = 2y • X = 2(30) • X = 60 • X + y = 90 • X + 30 = 90 • X = 60 Notice that the result for x is the same in both equations. The Closing LearningCheck Cool Down • Are there any misconceptions?