1. Solving Systems of Linear
Equations
Adding or Subtracting
File Name: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or Subtracting
Notes: 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 elimination
using 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 Try
Notes URL
- 3y
+ 3y
3x 21

12. You Try
3x - 4y = - 13
- 3x - 4y = - 67

13. What is the first step when solving with
elimination?
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 Elimination
5x + 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 = 22
3x - 4y = - 6 3x - 4y = - 6 Solution (2, 3)
-4x - 5y = - 17 -4x - 5y = - 17
4x - 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 18
and whose difference 22.
1. 14 and 4
2. 20 and -2
3. 24 and -6
4. 30 and 8

23. Solving Systems of Linear
Equations
Multiplication
File Name: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or Subtracting
Notes: F:TeachingNorth East Carolina Prep SchoolLesson PlansMathAssigments8 -- Solving Systems of Linear Equations by Adding or Subtracting Notes to PPT

24. 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.

25. 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

26. 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…

27. Solving a system of equations by elimination
using 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.

28. 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)

29. 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

30. 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.

31. x + 3y = 0 Write the equation in
5x + 9y = 12 Standard Form

32. x + 3y = 0 Determine which
5x + 9y = 12 variable to eliminate
Eliminate y
Eliminate x

33. x + 3y = 0 Multiply by the LCM
5x + 9y = 12
LCM is (-3)
WHY

34. x + 3y = 0 Multiply the Equation
5x + 9y = 12
-3(x + 3y = 0)
-3x – 9y = 0 5x + 9y = 12
5x + 9y = 12

35. -3x – 9y = 0 Solve using new
5x + 9y = 12 Equation

36. Add or Subtract to
cancel

37. Plug x back into the
UNCHANGED
Equation

38. YOU TRY
3x + 2y = 9 4x + 3y = 49
-6x – y = 0 12x + 3y = 129
4x + 2y = -44 5x + 4y = 25
4x – 8y = 16 4x + 12y = 108

39. 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)

40. 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

41. 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

42. What is the first step when solving with
elimination?
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.

43. 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

44. 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.

45. 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

46. 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

47. What is the best number to multiply the top
equation 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

48. Solve using elimination.
2x – 3y = 1
x + 2y = -3
1. (2, 1)
2. (1, -2)
3. (5, 3)
4. (-1, -1)

49. Find two numbers whose sum is 18
and whose difference 22.
1. 14 and 4
2. 20 and -2
3. 24 and -6
4. 30 and 8

50. 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