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# 8 - solving systems of linear equations by adding or subtracting

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### 8 - solving systems of linear equations by adding or subtracting

1. 1. Solving Systems of Linear Equations Adding or SubtractingFile Name: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or SubtractingNotes: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or Subtracting Notes to PPT
2. 2. Objective The student will be able to: 8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection
3. 3. Solving Systems of Equations  So far, we have solved systems using graphing and tabular method. These notes show how to solve the system algebraically using ELIMINATION with addition and subtraction.  Elimination is easiest when the equations are in standard form.
4. 4. Solving a system of equations by eliminationusing addition and subtraction. Step 1: Put the equations in Standard Form. Standard Form: Ax + By = C Step 2: Determine which Look for variables that have the variable to eliminate. same coefficient. Step 3: Add or subtract the Solve for the variable. equations. Step 4: Plug back in to find Substitute the value of the variable the other variable. into the equation. Step 5: Check your Substitute your ordered pair into solution. BOTH equations.
5. 5. 1) Solve the system using elimination. x+y=5 3x – y = 7 Step 1: Put the equations in They already are! Standard Form. Step 2: Determine which The y’s have the same variable to eliminate. coefficient. Add to eliminate y. Step 3: Add or subtract the x+ y=5 equations. (+) 3x – y = 7 4x = 12 x=3
6. 6. 1) Solve the system using elimination. x+y=5 3x – y = 7 x+y=5 Step 4: Plug back in to find the other variable. (3) + y = 5 y=2 (3, 2) Step 5: Check your solution. (3) + (2) = 5 3(3) - (2) = 7 The solution is (3, 2). What do you think the answer would be if you solved using substitution?
7. 7. 2) Solve the system using elimination. 5x + y = 9 5x – y = 1 Step 1: Put the equations in They already are! Standard Form. Step 2: Determine which The y’s have the same variable to eliminate. coefficient. Add to eliminate y. Step 3: Add or subtract the 5x + y = 9 equations. 5x – y = 1 10x = 10 x=1
8. 8. 2) Solve the system using elimination. 5x + y = 9 5x – y = 1 5x - y = 1 Step 4: Plug back in to find the other variable. 5(1) – y = 1 y=4 (1, 4) Step 5: Check your solution. 5(1) - (4) = 1 5(1) - (4) = 1 The solution is (1, 4). What do you think the answer would be if you solved using substitution?
9. 9. 3) Solve the system using elimination. -2x - 4y = 10 3x + 4y = - 3 Step 1: Put the equations in They already are! Standard Form. Step 2: Determine which The y’s have the same variable to eliminate. coefficient. Add to eliminate y. Step 3: Add or subtract the -2x - 4y = 10 equations. 3x + 4y = -3 x =7 x=7
10. 10. 3) Solve the system using elimination. -2x - 4y = 10 3x + 4y = - 3 3x + 4y = -3 3(7) + 4y = -3 Step 4: Plug back in to find 21 + 4y = -3 the other variable. 4y = -3 – 21 4y = -24 y = -6 Step 5: Check your solution. (7, -6) -2(7) - 4(-6) = 10 -14 – (-24) = 10 The solution is (7, -6). + 4(-6) = -3 3(7) 21 + (-24) = -3
11. 11. You TryNotes URL - 3y + 3y 3x 21
12. 12. You Try 3x - 4y = - 13 - 3x - 4y = - 67
13. 13. What is the first step when solving withelimination? 1. Add or subtract the equations. 2. Plug numbers into the equation. 3. Solve for a variable. 4. Check your answer. 5. Determine which variable to eliminate. 6. Put the equations in standard form.
14. 14. Which step would eliminate a variable? 3x + y = 4 3x + 4y = 6 1. Isolate y in the first equation 2. Add the equations 3. Subtract the equations 4. Multiply the first equation by -4
15. 15. Solve using elimination. 2x – 3y = -2 x + 3y = 17 1. (2, 2) 2. (9, 3) 3. (4, 5) 4. (5, 4)
16. 16. You Try (Exit Ticket) Solve the following using Elimination Solve the following using Elimination5x + 3y = 15 5x + 3y = 15- 2x - 3y = 12 - 2x - 3y = 12 Solution (9, -10)3x - 4y = - 21 3x - 4y = - 21- 3x - y = - 9 - 3x - y = - 9 Solution (1, 6)5x + 4y = 22 5x + 4y = 223x - 4y = - 6 3x - 4y = - 6 Solution (2, 3)-4x - 5y = - 17 -4x - 5y = - 174x - 3y = 9 4x - 3y = 9 Solution (3, 1)
17. 17. HOMEWORK 8– Systems of Linear Equations Adding & Subtracting Solve by Elimination 1
18. 18. 3) Solve the system using elimination. y = 7 – 2x 4x + y = 5 Step 1: Put the equations in 2x + y = 7 Standard Form. 4x + y = 5 Step 2: Determine which The y’s have the same variable to eliminate. coefficient. Subtract to eliminate y. Step 3: Add or subtract the 2x + y = 7 equations. (-) 4x + y = 5 -2x = 2 x = -1
19. 19. 2) Solve the system using elimination. y = 7 – 2x 4x + y = 5 y = 7 – 2x Step 4: Plug back in to find y = 7 – 2(-1) the other variable. y=9 (-1, 9) Step 5: Check your solution. (9) = 7 – 2(-1) 4(-1) + (9) = 5
20. 20. Find two numbers whose sum is 18and whose difference 22. 1. 14 and 4 2. 20 and -2 3. 24 and -6 4. 30 and 8
21. 21. Solving Systems of Linear Equations MultiplicationFile Name: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or SubtractingNotes: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or Subtracting Notes to PPT
22. 22. Vocabulary  Standard Form: Ax + By = C where A, B, and C are real numbers and A and B are not both zero. (4x + 5y = 25 or 0.5x + (- 5y) = (-4.75)  Coefficient: number which multiplies a variable. (5x; Five is the coefficient)  Least Common Multiple: the smallest factor that is the multiple of two or more numbers.
23. 23. Objective The student will be able to: 8.EE.8a: Analyze and solve pairs of simultaneous linear equations. 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 8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection 8.EE.8c: Analyze and solve pairs of simultaneous linear equations. Solve real- world and mathematical problems leading to two linear equations in two variables
24. 24. Solving Systems of Equations  So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication.  What happens when the coefficients are not the same?  We multiply the equations to make them the same! You’ll see…
25. 25. Solving a system of equations by eliminationusing multiplication. Step 1: Put the equations in Standard Form. Standard Form: Ax + By = C Step 2: Determine which Look for variables that have the variable to eliminate. same coefficient. Step 3: Multiply the Solve for the variable. equations and solve. Step 4: Plug back in to find Substitute the value of the variable the other variable. into the equation. Step 5: Check your Substitute your ordered pair into solution. BOTH equations.
26. 26. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 Step 1: Put the equations in They already are! Standard Form. None of the coefficients are the same! Find the least common multiple Step 2: Determine which of each variable. variable to eliminate. LCM = 6x, LCM = 2y Which is easier to obtain? 2y (you only have to multiply the bottom equation by 2)
27. 27. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 Multiply the bottom equation by 2 2x + 2y = 6 2x + 2y = 6 Step 3: Multiply the (+) 6x – 2y = 10 equations and solve. (2)(3x – y = 5) 8x = 16 x=2 2(2) + 2y = 6 Step 4: Plug back in to find 4 + 2y = 6 the other variable. 2y = 2 y=1
28. 28. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 (2, 1) Step 5: Check your solution. 2(2) + 2(1) = 6 3(2) - (1) = 5 Solving with multiplication adds one more step to the elimination process.
29. 29. x + 3y = 0 Write the equation in5x + 9y = 12 Standard Form
30. 30. x + 3y = 0 Determine which5x + 9y = 12 variable to eliminate Eliminate y Eliminate x
31. 31. x + 3y = 0 Multiply by the LCM5x + 9y = 12 LCM is (-3) WHY
32. 32. x + 3y = 0 Multiply the Equation5x + 9y = 12 -3(x + 3y = 0)-3x – 9y = 0 5x + 9y = 125x + 9y = 12
33. 33. -3x – 9y = 0 Solve using new5x + 9y = 12 Equation
34. 34. Add or Subtract tocancel
35. 35. Plug x back into theUNCHANGEDEquation
36. 36. YOU TRY 3x + 2y = 9 4x + 3y = 49-6x – y = 0 12x + 3y = 1294x + 2y = -44 5x + 4y = 254x – 8y = 16 4x + 12y = 108
37. 37. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Step 1: Put the equations in They already are! Standard Form. Find the least common multiple of each variable. LCM = 4x, LCM = 12y Step 2: Determine which variable to eliminate. Which is easier to obtain? 4x (you only have to multiply the top equation by -4 to make them inverses)
38. 38. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Multiply the top equation by -4 (-4)(x + 4y = 7) -4x – 16y = -28 Step 3: Multiply the 4x – 3y = 9) (+) 4x – 3y = 9 equations and solve. -19y = -19 y=1 x + 4(1) = 7 Step 4: Plug back in to find the other variable. x+4=7 x=3
39. 39. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 (3, 1) Step 5: Check your solution. (3) + 4(1) = 7 4(3) - 3(1) = 9
40. 40. What is the first step when solving withelimination? 1. Add or subtract the equations. 2. Multiply the equations. 3. Plug numbers into the equation. 4. Solve for a variable. 5. Check your answer. 6. Determine which variable to eliminate. 7. Put the equations in standard form.
41. 41. Which variable is easier to eliminate? 3x + y = 4 4x + 4y = 6 1. x 2. y 3. 6 4. 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
42. 42. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Step 1: Put the equations in They already are! Standard Form. Find the least common multiple of each variable. Step 2: Determine which LCM = 12x, LCM = 12y variable to eliminate. Which is easier to obtain? Either! I’ll pick y because the signs are already opposite.
43. 43. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Multiply both equations (3)(3x + 4y = -1) 9x + 12y = -3 Step 3: Multiply the (4)(4x – 3y = 7) (+) 16x – 12y = 28 equations and solve. 25x = 25 x=1 3(1) + 4y = -1 Step 4: Plug back in to find 3 + 4y = -1 the other variable. 4y = -4 y = -1
44. 44. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 (1, -1) Step 5: Check your solution. 3(1) + 4(-1) = -1 4(1) - 3(-1) = 7
45. 45. What is the best number to multiply the topequation by to eliminate the x’s? 3x + y = 4 6x + 4y = 6 1. -4 2. -2 3. 2 4. 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
46. 46. Solve using elimination. 2x – 3y = 1 x + 2y = -3 1. (2, 1) 2. (1, -2) 3. (5, 3) 4. (-1, -1)
47. 47. Find two numbers whose sum is 18and whose difference 22. 1. 14 and 4 2. 20 and -2 3. 24 and -6 4. 30 and 8
48. 48. Resources  Systems of Linear Equations Solve By Elimination Multiplication Math Planet  http://www.mathplanet.com/education/algebra-1/systems-of-linear-equations-and-inequalities/the-elimination-method-for-solving-linear-systems  Systems of Linear Equations Solve By Elimination Multiplication 7-4  http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/study_guide/pdfs/alg1_pssg_G056.pdf  Systems of Linear Equations Solve by Elimination Multiplication VIDEO:  http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-234s.html  Systems of Linear Equations Solve By Elimination Multiplication 8-4:  http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/9-4/GlencoeSG8-4.pdf  Systems of Linear Equations Solve By Elimination Multiplication 8-4:  http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/9-4/GlencoePWS8-4.pdf  Exam View: Systems of Linear Equations Solve By Elimination Multiplication 9-4:  ttp://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/examviewweb/ev9-4.htm