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Solving Systems of Linear Equations By: Kathi Mercer kathi.wisinski-mercer@cr.k12.de.us

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To solve a system of linear equations means to find one pair of (x, y) values that satisfy both equations.

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The two methods we will use to solve systems are: Graphing Substitution You must learn both methods to be prepared for the graded assessment.

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This presentation will allow you to learn both methods. If you already know the first method, you can take the quiz under the graphing link and then move on to the next method. If you need to learn either method, click on the link and follow the steps. The quiz at the end of each method will allow you to self assess if the concept was learned. Extra practice links will also be provided if necessary.

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Choose your topic: Graphing:Graphing To solve by graphing, graph the two lines and use the TI 83 plus calculator to find the point of intersection. Graphing Quiz Substitution:Substitution To solve by substitution, you combine two equations with two variables into a single equation with one variable by substituting from one equation to the other. Substitution Quiz

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Solving by graphing using the TI 83 plus calculator. Remember to clear the memory in your calculator in order to turn off all plots and reset your viewing window.

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Step 1: Both linear equations need to be in the form of y as a function of x. Example: Given Subtract x from both sides Divide by 10 on both sides Given Subtract 16x from both sides

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The two equations are now in the form y as a function of x. y = -1.6x + 24 and y = -x + 18

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Step 2: On your graphing calculator, type both equations in y = and make the graph. The screens should look similar to the two below.

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Step 3: Determine the point of intersection on the graph. On the calculator, follow these steps: 1.Type 2 nd,then Calculate. 2.Choose 5: intersect. 3.This will take you back to the graph and prompt you to press enter when the first line is selected and enter again when the second line is selected. 4.Next, you will guess where the point of intersection is located with your cursor and press enter. 5.Finally, it will tell you x = 10 and y = 8.

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This is a video of the proper procedure to follow to obtain the answer of (10, 8). Click on the picture below to view the video.

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Quiz: Solve the system through graphing: Given: 2x + y = 5 and 4x – 3y = 10 1.Choose which option correctly states each equation in the form of y as a function of x. A.A.y = -2x +5 and y = B.B.y = 2x +5 and y = C.C.y = -2x +5 and y =

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Answer A is correct, you may move on to the next question. Question Two Question Two

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Your answer is incorrect, try again.try again For more practice visit: http://www.kutasoftware.com/free.html Under writing linear equations.

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2.Using the new equations, solve by graphing. The solution set is: A.A.(.5, 4) B.B.(4,.5) C.C.(4.5,.5)

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Answer A is correct, you may move on to the next method!next method

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Your answer is incorrect, try again.try again There are more practice problems at: http://www.kutasoftware.com/free.html Under solving systems of equations by graphing.

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Substitution: An algebraic method To solve by substitution, you “combine the equations with two variables into a single equation with one variable by substituting from one equation to the other” (Fey, Hart, Hirsch, Schoen, & Watkins, 2008, p. 51).

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Step 1: Solve one of the equations for x in terms of y. Example: Given: 16x + 10y = 240 and x + y = 18, take one of the equations and solve for x. x + y = 18 Subtract y from each side. - y -y x = -y + 18

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Step 2: Take the x = equation and substitute it’s value in for the x in the other equation. Example: Since x = -y + 18 and 16x + 10y = 240, Substitute –y + 18 in for x to obtain 16(-y + 18) + 10y = 240.

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Step 3: Solve the equation for y. Example: 16(-y + 18) + 10y = 240 Distribute the 16 -16y + 288 + 10y = 240 Add like terms -6y + 288 = 240 Solve for y - 288 -288 Subtract 288 from both sides -6y = -48 Divide both sides by -6 -6 -6 y = 8

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Step 4: Use y = 8 to solve for x in either equation. Example: We know y = 8 and x + y = 18. Therefore, x + 8 = 18 and we solve. X + 8 = 18 - 8 -8 Subtract 8 from both sides. x = 10

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Our answer is written in the form (x, y) as (10, 8). Take the Quiz

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Quiz Solve the system through substitution: Given: 3x + 2y = 30 and x + y = -60 1.Choose which option correctly states the equation x + y = -60 in the form of x as a function of y. A.A.x = y – 60 B.B.x = -y - 60 C.C.x = -y + 60

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Answer B is the correct answer. Please move on to the next question. Next Question

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Your answer is incorrect, try again.try again For more example problems visit: http://hotmath.com/help/gt/genericalg1/section_5_2.html http://hotmath.com/help/gt/genericalg1/section_5_2.html

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2.Using substitution, the solution set is: AA. (.5, 4) BB. (4,.5) CC. (4.5,.5)

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Correct, click on the final assessment which you will print and hand in at the end of class. Geometry Graded Assessment

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Your answer is incorrect, try again.try again For more example problems visit: http://hotmath.com/help/gt/genericalg1/section_5_2.html

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