Download Solving Systems of Equations using Substitution and Elimination

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 -10
y=mx+b
Step 2: Substitute new expression (y = 3x -10) for the variable
(y) in the second equation.
x + 4y = -1
x + (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 -10
y = 3(3) -10
y = 9 -10
y = -1
Note: The reason of solving systems of equations using
substation would be to eliminate graphing.
Ex1)
1) 2x + 2y = 7
2)-4x -y = 8
Ex2)
y = -3x + 12
5x + 4y = 27
Ex3)
y = -1/2x + 3
y = 4/3x – 8
Ex4)
y = .25x + 6.1
y = .15x + 8.2
Ex5)
y = 10x + 212
x + 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 = 5
2) 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 = -21
5x +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 = b
3b + 4c + 2o = 7.70
coke = c
4b + 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 = 10
3x + y + 4z = 12
x + 5y + 2z = 20
Ex2)
5a +1b +2c = 1.26
2a +3b +4c = 1.88
3a +4b +1c = 1.24
Ex3)
5x -2y +4z = 19
9x +3y -9z = 129
4x -4y +2z = 2
Ex4)
x +y -z = -1
4x -3y +2z = 16
2x -2y -3z = 5
Ex5)
3a + 25b +20c = 40
1a +1b +2c = 20
2a +5b +4c = 50
MINI QUIZ
1)
y = 5x +12
5 = y +7x
2)
10 = 2x +y
20 = -y +4x
3)
10x = y +24
y = 3x +4
4)
12x +9y = 3
2x +4y = -12
5)
-20x +3y = -22
-12x -7y = -66
6)
-7x +4y = -42
6x +9y = -51
7)
5a +3b +c = 24
6a +5b –c = 22
3a +4b +3c = 33
8)
7x +3y +6z = 16
4x +y +2z = 2
5x -5 -3z = -23
9)
-6a +5b +7c = -35
4a –b -3c = 23
-2a =b -4c = 6
Appendix
Mini Quiz
1)
5 = y +7x
5 = (5x +2) +7x
5 = 2 +12x
-2 +5 = 12x
3/12 = 12x/12
.25 = x
y = 5x +12
y = 5(.25) +12
y = 1.25 +12
y = 13.25
2)
10 = 2x +y
10 = 2x + (4x -20)
10 = 6x -20
20 +10 = 6x
30/6 = 6x/6
5=x
y = 4x -20
y = 4(5) -20
y = 20 -20
y=0
3)
10x = y +24
10x = (3x +4) +20
10x = 3x +28
-3x +10x = 28
7x/7 = 28/7
x=4
y = 3x +4
y = 3(4) +4
y = 12 +4
y = 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 = 24
6a +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 = -966
231a +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 = 16
3(5x -y -3z = -23)
7x +3y +6z = 16
15x -3y -9z = -69
--------------------22x -3z = -53
-------------------------------4x +y +2z = 2
5x -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 = -35
5(4a -b -3c = 23)
-6a +5a +7c = -35
20a -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 = 7
2)-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:
Step4:
2x + 3y = 7
2(3.1) + 3y = 7
2(3.1) + 3y = 7
6.2 + 3y = 7
3y = 7 -6.2
3y/3 = .8/3
y = .26
Ex2)
y = -3x + 12
5x + 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 = 27
5(3) + 4 = 27
Step4:
5(3) + 4y = 27
15 + 4y = 27
4y = 27 – 15
4y/4 = 12/4
y=3
Ex3)
y = -1/2x + 3
y = 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 + 3
y = -1/2(6) + 3
Step4:
y = -1/2(6) + 3
y = -3 +3
y=0
Ex4)
y = .25x + 6.1
y = .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/.1
x = 21
Step3:
y = .25x + 6.1
y = .25(21) + 6.1
Step4:
y = .25x + 6.1
y = 5.25 + 6.1
y = 11.35
Ex5)
y = 10x + 212
x + 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 + 212
y = 10(3) + 212
Step4:
y = 10(3) + 212
y = 30 + 212
y = 242
Elimination:
Ex1)
Step1:
3x -2y = 8
6x –y = 16
6(3x -2y = 8)
-12(6x -y = 16)
18x -12y = 48
-72x +12y = -192
---------------------54x/-54 = -144/-54
x = 2.6
Step2:
Ex2)
18x -12y = 48
18(2.6) -12y = 48
46.8 -12y = 48
-12y = 48 -46.8
-12y/-12 = -1.2/-12
y = -.1
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 = -21
6x +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 = 8
1.5x +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 = 266
35x +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 = 51
3x +4(6) = 51
3x +24 = 15
3x = 51 -24
3x/3 = 27/3
x=9
Solving 3x3:
Ex1)
x + y + 2z = 10
3x + y + 4z = 12
x + 5y + 2z = 20
Step2:
-(x + y + 2z = 10)
(x + 5y + 2z = 20)
-x - y - 2z = -10
x + 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 = 1.26
2a +3b +4c = 1.88
3a +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 = 19
9x +3y -9z = 129
4x -4y +2z = 2
Step2:
3(5x -2y +4z = 19)
2(9x +3y -9z = 129)
15x -6y +12z = 57
18x +6y -18z = 259
----------------------33x -6y = 315
------------------------------------4(9x +3y -9z = 129)
3(4x -4y +2z = 2)
36x +12y -36z = 516
12x -12y +6z = 6
----------------------48x -30z = 510
Step3:
-30(33x -6z = 315)
6(48x -30z = 510)
-990x +180z = -9450
288x -180z = 3060
-----------------------702x/-702 = -6390/-702
x=9
--------------------------------------33x -6z = 315
33(9) -6z = 315
297 -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 = -1
4x -3y +2z = 16
2x -2y -3z = 5
Step2:
-4(x +y -z = -1)
4x -3y +2z = 16
-4x -4y +4z = 4
4x -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 = 20
7(1y +8z = 6)
-7y +6z = 20
7y +56z = 42
---------------62z/62 = 62/62
z=1
1y +8z = 6
1y +8(1) = 6
1y +8 = 6
1y = 6 -8
y = -2
Step4:
Ex5)
Step2:
x +y –z = -1
x +(-2) –(1) = -1
x = -1 +1 +2
x=2
3a + 25b +20c = 40
1a +1b +2c = 20
2a +5b +4c = 50
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 = -40
2a +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:
http://school.discovery.com/homeworkhelp/webmath/solver2.html
http://mathforum.org/dr.math/tocs/linear.al.high.html
http://www.sosmath.com/index.html