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- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. You TryNotes URL - 3y + 3y 3x 21
- 12. You Try 3x - 4y = - 13 - 3x - 4y = - 67
- 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. 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. Solve using elimination. 2x – 3y = -2 x + 3y = 17 1. (2, 2) 2. (9, 3) 3. (4, 5) 4. (5, 4)
- 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. HOMEWORK 8– Systems of Linear Equations Adding & Subtracting Solve by Elimination 1
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. x + 3y = 0 Write the equation in5x + 9y = 12 Standard Form
- 30. x + 3y = 0 Determine which5x + 9y = 12 variable to eliminate Eliminate y Eliminate x
- 31. x + 3y = 0 Multiply by the LCM5x + 9y = 12 LCM is (-3) WHY
- 32. x + 3y = 0 Multiply the Equation5x + 9y = 12 -3(x + 3y = 0)-3x – 9y = 0 5x + 9y = 125x + 9y = 12
- 33. -3x – 9y = 0 Solve using new5x + 9y = 12 Equation
- 34. Add or Subtract tocancel
- 35. Plug x back into theUNCHANGEDEquation
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Solve using elimination. 2x – 3y = 1 x + 2y = -3 1. (2, 1) 2. (1, -2) 3. (5, 3) 4. (-1, -1)
- 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. 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

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