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Solving Systems of Equations using Substitution and

Elimination

This tutorial kit will teach you step by step how to solve systems of equations using substitution and elimination. At the end of the tutorial kit there will be a mini quiz so you can test out the new skills you have learned. There will also be a appendix at the end with the answers to the practice, and mini quiz questions. There will also be a few addresses of some helpful web sites.

Solving 2x2 Systems

In a 2x2 equation you will have two different equations that both contain two unknowns (x, y)

Solving Using Substation

You use substation when one of the equations has a 1x, or 1y. In this case you could use either.

Ex) 1) x + 4y = -1 y = 3x -10

Step 1: Put one of the equations into y = m x + b form.y = 3x -10y = m x + b

Step 2: Substitute new expression (y = 3x -10) for the variable (y) in the second equation.

x + 4y = -1x + (3x -10) = -1

Step 3: Once you have your new equation you can now solve for the unknown variable (in this case “x”)

x + 4(3x -10) = -1 x + 12x -40 = -1 13x -40 = -1 13x = 40 -1 13x/13 = 39/13 x = 3

Step 4: Now that you know what x equals, substitute the number into one of the original equations.

y = 3x -10y = 3(3) -10y = 9 -10y = -1

Note: The reason of solving systems of equations using substation would be to eliminate graphing.

Ex1) 1) 2x + 2y = 72)-4x -y = 8

Ex2) y = -3x + 125x + 4y = 27

Ex3) y = -1/2x + 3y = 4/3x – 8

Ex4) y = .25x + 6.1y = .15x + 8.2

Ex5) y = 10x + 212x + y = 245

Solving using Elimination

You use elimination when both equations have a coefficient in front of the unknown variable (ex. 4y)

Ex) 1) 3x +4y = 52) 5x -4y = -13

Step 1: Check to see if you have the same coefficient in front of both sets of the x or y variables. (One must be a positive, and the other must be a negative) If they are not the same number you will have to multiply one, or both of the equations by a number so that you will be able cancel the numbers. In this question you don’t have to do this.

The first thing you do now, is add the two equations together. (one set of variables must cancel out.

3x +4y = 5 5x -4y = -13

--------------- 8x/8 = -8/8

x = -1

Step 2: Substitute this number into either one of the original equations to solve for the other variable (y)

3x +4y = 5 3(-1) +4y = 5

-3 +4y = 5 -3 +4y = 5 4y = 3 +5 4y/4 = 8/4 y = 2

Note: The reason for solving systems of equations using elimination is to eliminate graphing.

Ex1) 3x -2y = 8 6x –y = 16

Ex2) 6x +3y = -215x +5y = -25

Ex3) .25x +.75y = .6.15x +.35y = .8

Ex4) 5x +7y = -38 7x +10y = -54

Ex5) 3x +4y = 51 -6x +7y = -12

Solving 3x3 Systems

3x3 equations are equations that have 3 unknown variables. They can be word equations.

Ex) 2 cans of beans, 2 bottles of coke, and 1 glass of orange juice $4.20. 3 cans of beans, 4 bottles of coke and 2 glasses of orange juice costs $7.70. 4 cans of beans, 3 bottles of coke and 5 glasses of orange juice costs $9.80.

Step 1: Read the question and figure out what equation it is going to be…. NOTE: When solving the problem you must use EVERY equation.

2b + 2c + 1o = 4.20 beans = b3b + 4c + 2o = 7.70 coke = c4b + 3c + 5o = 9.80 orange juice = o

Step 2: Choose 2 equations and solve for the unknown variables, just like in 2x2 equations.

-2(2b + 2c + 1o = 4.20) 3b + 4c + 2o = 7.70

-4b - 4c - 2o = -8.40 3b +4c + 2o = 7.7o------------------------

-1b/-1 = -.7/-1 b = .7

Step 3: Now use a different equation and find another one of the unknown variables.

3(2b + 2c + 1o = 4.20) -2(4b + 3c + 5o = 9.80)

6b + 6c + 3o = 12.6 -8b - 6c - 10o = -19.6

------------------------ -2b -7o = -7 (now you can substitute the variable you -2(.7) -7o = -7 have already found) -1.4 -7o = -7

-7o = -7 + 1.4 -7o/-7 = -5.6/-7

o = .8

Step 4: Pick one of the original equations and fill in the values of the two unknown variables you found, and solve for the last one.

2b + 2c +1o = 4.20 2(.7) + 2c + 1(.8) = 4.20 1.4 + 2c + .8 = 4.20

2c = 4.20 -.8 -1.4 2c/2 = 2/2 c = 1

NOTE: To check to see if you got it all correct, plug the numbers you found into the equations and see if it equal the other side

Ex1) x + y + 2z = 103x + y + 4z = 12x + 5y + 2z = 20

Ex2) 5a +1b +2c = 1.26

2a +3b +4c = 1.883a +4b +1c = 1.24

Ex3) 5x -2y +4z = 199x +3y -9z = 1294x -4y +2z = 2

Ex4) x +y -z = -14x -3y +2z = 16 2x -2y -3z = 5

Ex5) 3a + 25b +20c = 401a +1b +2c = 202a +5b +4c = 50

MINI QUIZ

1) y = 5x +125 = y +7x

2) 10 = 2x +y20 = -y +4x

3) 10x = y +24y = 3x +4

4) 12x +9y = 32x +4y = -12

5) -20x +3y = -22-12x -7y = -66

6) -7x +4y = -426x +9y = -51

7) 5a +3b +c = 246a +5b –c = 223a +4b +3c = 33

8) 7x +3y +6z = 164x +y +2z = 25x -5 -3z = -23

9) -6a +5b +7c = -354a –b -3c = 23-2a =b -4c = 6

Appendix

Mini Quiz

1) 5 = y +7x5 = (5x +2) +7x5 = 2 +12x

-2 +5 = 12x 3/12 = 12x/12 .25 = x

y = 5x +12y = 5(.25) +12y = 1.25 +12y = 13.25

2) 10 = 2x +y10 = 2x + (4x -20)10 = 6x -20

20 +10 = 6x 30/6 = 6x/6

5 = x

y = 4x -20y = 4(5) -20y = 20 -20y = 0

3) 10x = y +2410x = (3x +4) +2010x = 3x +28

-3x +10x = 28 7x/7 = 28/7

x = 4

y = 3x +4y = 3(4) +4y = 12 +4y = 16

4) 12x +9y = 3 -6(2x +4y = -12)

12x +9y = 3 -12x -24y = 72 ----------------- -15y/-15 = 75/-15

y = -5

12x +9y = 3 12x +9(-5) = 3 12x -45 = 3

12x = 45 +3 12x/12 = 48/12 x = 4

5) 7(-20x +3y = -22)3(-12x -7y = -66)

-140x +21y = -154 -36x -21 = -198----------------------

-176x/-176 = -352/-176 x = 2

-20x +3y = -22 -20(2) +3y = -22

-40 +3y = -22 3y = -22 +40 3y/3 = 18/3 y = 6

6) 6(-7x +4y = -42)7(6x +9y = -15)

-42x +24y = -252 42x +63y = -357--------------------- 87y/87 = -609/87 y = -7

6x +9y = -51

6x +9(-7) = -51 6x -63 = -51 6x = -51 +63 6x/6 = 12/6 x = 2

7) 5a +3b +c = 246a +5b -c = 22------------------ 11a +8b = 46

----------------------------------3(6a +5b -c = 22)(3a +4b +3c = 33)

18a +15b -3c = 66 3a +4b +3c = 33--------------------- 21a +19b = 99

------------------------------------21(11a +8b = 46)11(21a +19b = 99)

-231a -168b = -966231a +209b = 1089----------------------- 41b/41 = 123/41 b = 3

11a +8b = 46 11a +8(3) = 46

11a +24 = 46 11a = 46 -24 11a/11 = 22/11 a = 2

5a +3b +c = 24 5(2) +3(3) +c = 24

10 +9 +c = 24 c = 24 -10 -9 c = 5

8) 7x +3y +6z = 163(5x -y -3z = -23)

7x +3y +6z = 1615x -3y -9z = -69-------------------- -22x -3z = -53

--------------------------------4x +y +2z = 25x -y -3z = -23------------------9x -1z = -21

--------------------------------22x -3z = -53-3(9x -12 = -21)

22x -3z = -53-27x +3z = 63----------------- -5x/-5 = 10/-5 x = -2

22x -3z = -53 22(-2) -3z = -53

-44 -3z = -53 -3z = -53 +44 -3z/-3 = -9/-3

z = 3

7x +3y +6z = 16 7(-2) +3y +6(3) = 16 -14 +3y +18 = 16

3y = 16 -18 +14 3y/3 = 12/3 y = 4

9) -6a +5b +7c = -355(4a -b -3c = 23)

-6a +5a +7c = -3520a -5b -15c = 115----------------------- 14a -8c = 80

----------------------------------- 4a -b -3c = 23-2a +b -4c = 6------------------ 2a -7c = 29

-----------------------------------14a -8c = 80

-7(2a -7c = 29)

14a -8c = 80 -14a +49c = -203

----------------41c/41 = -123/41 c = -3

14a -8c = 80 14a -8(-3) = 80 14a +24 = 80

14a = 80 -24 14a/14 = 56/14 a = 4

-6a +5b +7c = -35 -6(4) +5b +7(-3) = -35

-24 +5b -21 = -35 5b = -35 +24 +21

5b/5 = 10/5 b = 2

Substitution:

Ex1) 1) 2x + 2y = 72)-4x -y = 8

Step1: -4x -y = 8-8 -4x = y -y = -4x -8

Step2: 2x + 3y = 7 2x + 3(-4x -8) = 7

2x -12x -24 = 7 -10x -24 = 7

-10x = 24 + 7 -10x/-10 = 31/-10

x = 3.1

Step3: 2x + 3y = 7 2(3.1) + 3y = 7

Step4: 2(3.1) + 3y = 7 6.2 + 3y = 7

3y = 7 -6.2 3y/3 = .8/3 y = .26

Ex2) y = -3x + 125x + 4y = 27

Step1: y = -3x + 12

Step2: 5x + 4y = 27 5x + 4(-3x + 12) = 27

5x -12 + 48 = 27 -7x + 48 = 27

-7x = -48 + 27 -7x/-7 = -21/-7 x = 3

Step3: 5x + 4 = 275(3) + 4 = 27

Step4: 5(3) + 4y = 27 15 + 4y = 27 4y = 27 – 15 4y/4 = 12/4 y = 3

Ex3) y = -1/2x + 3y = 4/3x – 8

Step1: y = 4/3x -8

Step2: y = -1/2x -8 (4/3x -8) = -1/2x + 3

-8 -3 = -4/3x -1/2x -11 = -4/3x -1/2x

-11 = -8/6x -3/6x -11/-11/6 = -11/6x/-11/6

6 = x

Step3: y = -1/2x + 3y = -1/2(6) + 3

Step4: y = -1/2(6) + 3y = -3 +3y = 0

Ex4) y = .25x + 6.1y = .15x + 8.2

Step1: y = .25x + 6.1

Step2: y = .15x + 8.2 (.25x + 6.1) = .15x + 8.2 .25x -.15 = -6.1 + 8.2

.1x/.1 = 2.1/.1x = 21

Step3: y = .25x + 6.1y = .25(21) + 6.1

Step4: y = .25x + 6.1y = 5.25 + 6.1y = 11.35

Ex5) y = 10x + 212x + y = 245

Step1: y = 10x + 212

Step2: x + y = 245

x + (10x + 212) = 245 x + 10x = 245 -212 11x/11 = 33/11

x = 3

Step3: y = 10x + 212y = 10(3) + 212

Step4: y = 10(3) + 212y = 30 + 212y = 242

Elimination:

Ex1) 3x -2y = 8 6x –y = 16

Step1: 6(3x -2y = 8) -12(6x -y = 16)

18x -12y = 48 -72x +12y = -192 ---------------------

-54x/-54 = -144/-54 x = 2.6

Step2: 18x -12y = 48 18(2.6) -12y = 48

46.8 -12y = 48 -12y = 48 -46.8 -12y/-12 = -1.2/-12 y = -.1

Ex2) 6x +3y = -21

5x +5y = -25

Step1: 6x +3y = -21 -1.2(5x +5y = -25)

6x +3y = -21-6x -6y = 30----------------3y/-3 = -9/-3 y = 3

Step2: 6x +3y = -216x +3(-3) = -21 6x -9 = -21 6x = -21 +9 6x/6 = -12/6 x = -2

Ex3) .25x +.75y = .6.15x +.35y = .8

Step1: -6(.25x +.75y = .6)10(.15x +.35y = .8)

-1.5x -4.4y = -3.6 1.5x +3.5y = 8-------------------1y/-1 = 4.4/-1 y = -4.4

Step2: 1.5x +3.5y = 81.5x +3.5(-4.4) = 8 1.5x -15.4 = 8 1.5x = 8 +15.4 1.5x/1.5 = 23.4/1.5

x = 15.6

Ex4) 5x +7y = -38 7x +10y = -54

Step1: -7(5x +7y = -38)5(7x +10y = -54)

-35x -49y = 26635x +50z = -270-------------------- z = -4

Step2: 7x +109 = -54 7x +10(-4) = -54

7x -40 = -54 7x = -54 +40 7x/7 = -14/7 x = -2

Ex5) 3x +4y = 51 -6x +7y = -12

Step1: 2(3x +4y = 51)-6x +7y = -12

6x +8y = 102-6x +7y = -2

----------------15y/15 = 90/15 y = 6

Step2: 3x +4y = 513x +4(6) = 51 3x +24 = 15

3x = 51 -24 3x/3 = 27/3 x = 9

Solving 3x3:

Ex1) x + y + 2z = 103x + y + 4z = 12x + 5y + 2z = 20

Step2: -(x + y + 2z = 10)

(x + 5y + 2z = 20)

-x - y - 2z = -10x + 5y + 2z = 20-------------------- 4y/4 = 10/4

y = 2.5

Step3: -2(x + y + 2z = 10) 3x + y + 4z = 12

-2x - 2y - 4z = -20 3x + y + 4z = 12

---------------------- x -1y = -8

x -1(2.5) = -8 x - 2.5 = -8 x = -8 +2.5 x = -5.5

Step4: x + y + 2z = 10(-5.5)+(2.5)+2z = 10

2z = 10 +5.5 -2.5

2z/2 = 13/2 z = 6.5

Ex2) 5a +1b +2c = 1.262a +3b +4c = 1.883a +4b +1c = 1.24

Step2: -2(5a +1b +2c = 1.26) 2a + 3b +4c = 1.88

-10a -2b -4c = -2.52 2a +3b +4c = 1.88------------------------ -8a +1b = -.64

-----------------------------------2a +3b +4c = 1.88

-4(3a +4b +1c = 1.24)

2a +3b +4c = 1.88 -12a -16b -4c = -4.96

---------------------------10a -13b = -3.08

Step3: 13(-8a +1b = -.64) -10a -13b = -3.08

-104a +13b = -8.32-10a -13b = -3.08-----------------------114a/-114 = -11.4/-114

a = .1--------------------------------------

-8a + 1b = -.64 -8(.1) + 1b = -.64

-.8 +1b = -.64

b = -.64 +.8 b = .16

Step 4: 5a +1b +2c = 1.26 5(.1) +1(.16) +2c = 1.26

.5 + .16 +2c = 1.26 2c = 1.26 - .5 - .16 2c/2 = .6/2 c = .3

Ex3) 5x -2y +4z = 199x +3y -9z = 1294x -4y +2z = 2

Step2: 3(5x -2y +4z = 19)2(9x +3y -9z = 129)

15x -6y +12z = 5718x +6y -18z = 259-----------------------

33x -6y = 315-------------------------------------

4(9x +3y -9z = 129)3(4x -4y +2z = 2)

36x +12y -36z = 51612x -12y +6z = 6-----------------------

48x -30z = 510

Step3: -30(33x -6z = 315) 6(48x -30z = 510)

-990x +180z = -9450288x -180z = 3060

------------------------702x/-702 = -6390/-702 x = 9

---------------------------------------33x -6z = 315

33(9) -6z = 315297 -6z = 315 -6z = 315 -297 -6z/-6 = 18/-6 z = -3

Step4: 5x -2y +4z = 19 5(9) -2y +4(-3) = 19

45 -2y -12 = 19 -2y = 19 -45 +12 -2y/-2 = -14/-2 y = 7

Ex4) x +y -z = -14x -3y +2z = 16 2x -2y -3z = 5

Step2: -4(x +y -z = -1)4x -3y +2z = 16

-4x -4y +4z = 44x -3y +2z = 16

--------------------- -7y +6z = 20

--------------------------------------4x -3y +2z = 16

-2(2x -2y -3z = 5)

4x -3y +2z = 16 -4x +4y +6z = -10

-----------------------1y +8z = 6

Step3: -7y +6z = 207(1y +8z = 6)

-7y +6z = 207y +56z = 42----------------62z/62 = 62/62

z = 1

1y +8z = 6 1y +8(1) = 6

1y +8 = 6 1y = 6 -8 y = -2

Step4: x +y –z = -1 x +(-2) –(1) = -1

x = -1 +1 +2 x = 2

Ex5) 3a + 25b +20c = 401a +1b +2c = 202a +5b +4c = 50

Step2: 3a +25b +20c = 40 -3( 1a +1b +2c = 20)

3a +25b +20c = 40-3a -3b -6c = -6o

------------------------22b +14c = -20

--------------------------------------

-2(1a +1b +2c = 20)2a +5b +4c = 50

-2a -2b -4c = -402a +5b +4c = 50-------------------3b/3 = 10/3 b = 3.33

Step3: 22b +14c = -20 22(3.33) + 14c = -20

73.33 +14c = -20 14c = -20 -73.33 14c/14 = -93.33/14 c = -6.67

Step4: 3a +25b +20c = 40 3a +25(3.33) +20(-6.67) = 40

3a +83.25 -133.4 = 40 3a = 40 -83.25 +133.4 3a/3 = 90.15/3 a = 30.05

Web Sites To Help You:

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http://mathforum.org/dr.math/tocs/linear.al.high.html

http://www.sosmath.com/index.html