Module 4 - Case SYSTEMS OF EQUATIONS AND INEQUALITIES Assignment Expectations Complete the following problems using the Case 4 Answer Template. 1.What are the 3 methods for solving systems of equations? Which one do you prefer and why? 2.Describe a consistent system. 3.Describe an inconsistent system. 4.Describe a dependent system. Solve the following systems of equations using the graphing method. What type of system is it? Name the solution if there is one. 5. x + y = 5 -x + y = 3 6. 2x - 2y = 8 x-y=4 7. 3x - y = -2 3x - y = 4 8. y = 6 x = -3 Solve the following systems of equations using the substitution method. What type of system is it? Name the solution if there is one. 9. x = 6y + 2 3x - 18y = 4 10. x - y = 3 x + 3y = 6 11. y = -2x 4x - 3y = 12 12. 2x - 5y = 15 x - 7y = 3 Solve the following systems of equations using the addition (elimination) method. What type of system is it? Name the solution if there is one. 13. x + y = 6 x-y=4 14. -2x + 3y = 5 2x - y = 1 15. 5x + y = 2 3x + y = 1 16. 5x = 20 + y 16 = 3y + 4x Solve the following systems of inequalities using the graphing method. Shade the solution set. 17. x + y > 4 y + 3x < 6 18. x 5 y<3 19. 3x - 2y> 8 -2y + 3x < 12 20. 2x + y < 8 x4 MAT101 Case 4 Answer Template I affirm that this assignment contains all original work. I am familiar with Trident University's Academic Integrity policy located in the Trident Policy Handbook. I affirm that I have not engaged in direct duplication, copy/pasting, sharing assignments, collaboration with others, contract cheating and/or obtaining answers online, paraphrasing, or submitting/facilitating the submission of prior work. Work found to be unoriginal and in violation of this policy is subject to consequences such as a failing grade on the assignment, a failing grade in the course, and/or elevated academic sanctions. My name below affirms that the assignment was completed individually, and all work presented is my own. Name: Date: 20150610 Instructor: Use this template to insert your answers for the assignment. Please use one of the four methods for showing your work (EE, Math Type, ALT keys, or neatly typed). Remember that your work should be clear and legible. Problems need to include all required steps and answer(s) for full credit. All answers need to be reduced to lowest terms where possible. 1. What are the 3 methods for solving systems of equations? Which one do you prefer and why? Answer: The three methods are elimination (addition), graphing and substitution. All three methods will get the same answer. Preference/reason: I prefer elimination because I am terrible at math and elimination is easy for me when I just don't know what else to do. 2. Describe a consistent system. Answer: a consistent system has at least one solution. 3. Describe an inconsistent system. Answer: An inconsistent system has no solution _________________ 4. Describe a dependent system. Answer: A dependent system means that all points that work in one of them also work in the other one. On a graph, one line is lying entirely on top of the other one, so when both are graphed, you would really see only one line on the graph, because they are imposed on top of each other. One of them totally depends on the other one. Go to 'insert' and 'shapes'. Choose the circles for the points and the lines to connect the dots. Use the arrows to represent the shading. For example: Solve the following systems of equations using the graphing method. What type of system is it? Name the solution if there is one. 5. Calculations: Step 1: Solve x + y = 5 for x: x + y + y = 5 + y (Add -y to both sides) x = y + 5 Step 2: Substitute (y+5) for x in -x + y = 3: x + y = 3 (y + 5) + y = 3 2y - 5 =3 (Simplify both sides of the equation) 2y - 5 + 5 = 3 + 5 (Add 5 to both sides) 2y = 8 2y / 2 = 8 / 2 (Divide both sides by 2) y=4 Step 2: Substitute (4) for y in x = y + 5: X = y + 5 X = 4 + 5 X = 1(Simplify both sides of the equation) Answer: _____ x=1 and y=4_____________ 6. Calculations: Step 1: Solve x - y = 4 for x: x - y + y = 4 + y (Add y to both sides) x=y+4 Step 2: Substitute (y+4) for x in 2x 2y = 8 2x 2y = 8 2(y+4) 2y = 8 8 = 8 (Simplify both sides of the equation) 8 + 8 = 8 + 8 (Add -8 to both sides) 0=0 Answer: ___Infinitely many solutions__________________ 7. Calculations: Step 1: Solve 3x - y = 2 for y: 3x - y + 3x = 2 + 3x (Add -3x to both sides) y = 3x 2 y / 1 = 3x2 / 1 (Divide both sides by -1) y = 3x + 2 Step 2: Substitute (3x + 2) for y in 3x - y =4 3x - y = 4 3x (3x+2) = 4 2 = 4 (Simplify both sides of the equation) 2 + 2 = 4 + 2 (Add 2 to both sides) 0=6 Answer: _ No solutions___________ 8. Calculations: Answer: _________________________________________________ Solve the following systems of equations using the substitution method. What type of system is it? Name the solution if there is one. 9. Calculations: Step 1: Solve x = 6y + 2 for x: Step2: Substitute (6y+2) for x in 3x 18y = 4 3x 18y = 4 3(6y+2) 18y = 4 6 = 4 (Simplify both sides of the equation) 6 + 6 = 4 + 6 (Add -6 to both sides) 0 = 2 Answer: ____No Solution______ 10. Calculations: Step 1: Solve x - y = 3 for x x - y + y = 3 + y (Add y to both sides) x=y+3 Step2: Substitute (y+3) for x in x + 3y = 6 x + 3y = 6 y + 3 + 3y = 6 4y + 3 = 6 (Simplify both sides of the equation) 4y + 3 + 3 = 6 + 3 (Add -3 to both sides) 4y = 3 4y / 4 = 3 / 4 (Divide both sides by 4) y=3/4 Step2: Substitute (3 / 4) for y in x = y + 3 x=y+3 x=3/4+3 x = 15 / 4 (Simplify both sides of the equation) Answer: __ x = 15 / 4 and y = 3 / 4___ 11. Calculations: Step 1: Solve y = 2x for y Step 2: Substitute (2x) for y in 4x 3y = 12 4x 3y = 12 4x - 3 2x = 12 10x = 12 (Simplify both sides of the equation) 10x / 10 = 12 / 10 (Divide both sides by 10) x=6/5 Step 3: Substitute (6 / 5) for x in y = 2x y = 2x y = 2(6 / 5) y = 12 / 5 (Simplify both sides of the equation) Answer: y = 12 / 5 and x = 6 / 5___ 12. Calculations: Step 1: Solve x 7y = 3 for x x 7y + 7y = 3 + 7y (Add 7y to both sides) x = 7y + 3 Step 2: Substitute (7y+3) for x in 2x 5y = 15 2x 5y = 15 2(7y+3) 5y = 15 9y + 6 = 15 (Simplify both sides of the equation) 9y + 6 + 6 = 15 + 6 (Add -6 to both sides) 9y = 9 9y / 9 = 9 / 9 (Divide both sides by 9) y=1 Step 3: Substitute (1) for y in x = 7y + 3 x = 7y + 3 x = (7)(1) + 3 x = 10 (Simplify both sides of the equation) Answer: ____ x=10 and y=1______ Solve the following systems of equations using the addition (elimination) method. What type of system is it? Name the solution if there is one. 13. Calculations: Step 1: Solve x + y = 6 for x x + y + y = 6 + y (Add -y to both sides) x = y + 6 Step 2: Substitute (y + 6) for x in x - y = 4 x-y=4 y + 6 - y = 4 2y + 6 = 4 (Simplify both sides of the equation) 2y + 6 + 6 = 4 + 6 (Add -6 to both sides) 2y = 2 2y / 2 = 2 / 2 (Divide both sides by -2) y=1 Step 2: Substitute (1) for y in x = y + 6 x = y + 6 x = 1 + 6 x = 5 (Simplify both sides of the equation) Answer: __ x=5 and y=1______ 14. Calculations: Step 1: Solve 2x - y = 1 for y 2x - y + 2x = 1 + 2x (Add -2x to both sides) y = 2x + 1 y / 1 = 2x + 1 / 1 (Divide both sides by -1) y = 2x 1 Step 2: Substitute (2x1) for y in 2x +3y = 5 2x + 3y = 5 2x +3 (2x1) = 5 4x - 3 = 5 (Simplify both sides of the equation) 4x - 3 + 3 = 5 + 3 (Add 3 to both sides) 4x = 8 4x / 4 = 8 / 4 (Divide both sides by 4) x=2 Step 2: Substitute (2) for x in y = 2x 1 y = 2x 1 y = (2)(2) 1 y = 3(Simplify both sides of the equation) Answer: ___ y=3 and x=2_____ 15. Calculations: Step 1: Solve 5x + y = 2 for y 5x + y + 5x =2 + 5x (Add -5x to both sides) y = 5x + 2 Step 2: Substitute (5x+2) for y in 3x + y = 1 3x + y = 1 3x + 5x + 2 = 1 2x + 2 = 1 (Simplify both sides of the equation) 2x + 2 + 2 = 1 + 2 (Add -2 to both sides) 2x = 1 2x / 2 = 1 / 2 (Divide both sides by -2) x=1/2 Step 3: Substitute (1 / 2) for x in y = 5x + 2 y = 5x + 2 y = 5(1 / 2) + 2 y = 1 / 2 (Simplify both sides of the equation) Answer: ___ y = 1 / 2 and x = 1 / 2________ 16. Calculations: Step 1: Solve 5x = y + 20 for y 5x + y = y + 20 + y (Add -y to both sides) 5x - y = 20 5x - y + 5x = 20 + 5x (Add -5x to both sides) y = 5x + 20 y / 1 = 5x + 20 / 1 (Divide both sides by -1) y = 5x 20 Step 2: Substitute (5x20) for y in 16 = 4x + 3y 16 = 4x + 3y 16 = 4x +3 (5x20) 16 = 19x - 60 (Simplify both sides of the equation) 16 + 19x = 19x - 60 + 19x (Add -19x to both sides) 19x + 16 = 60 19x + 16 + 16 = 60 + 16 (Add -16 to both sides) 19x = 76 19x / 19 = 76 / 19 (Divide both sides by -19) x=4 Step 3: Substitute (4) for x in y = 5x 20 y = 5x 20 y = (5)(4) 20 y = 0 (Simplify both sides of the equation) Answer: ___ y = 0 and x = 4_____ Solve the following systems of inequalities using the graphing method. Shade the solution set. 17. Calculations: Answer: _________________________________________________ 18. Calculations: Answer: _________________________________________________ 19. Calculations: Answer: _________________________________________________ 20. Calculations: Answer: _________________________________________________