Solving Systems of Equations

1
Solving Systems of Equations

• Cramers Rule
• with a 3 x 3 system
• By
• Jeffrey Bivin
• Lake Zurich High School
• jeff.bivin_at_lz95.org

Last Updated October 11, 2005
2
4x 3y z -10x 12y 2z -5x 18y z
4
System to be solved
jeff.bivin_at_lz95.org
3
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
Use determinants
X
jeff.bivin_at_lz95.org
4
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
5
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
6
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
1
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
7
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
1
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
8
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
9
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
10
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
11
4x 3y z -10x 12y 2z -5x 18y z
4
1
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
12
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
13
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
1
x
y
z
Fill the denominator's determinant with
the coefficients
X
x
y
z
jeff.bivin_at_lz95.org
14
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
X
x
y
z
jeff.bivin_at_lz95.org
15
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Now place the constants in the x column of
the numerator
X
x
y
z
jeff.bivin_at_lz95.org
16
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Now place the constants in the x column of
the numerator
X
x
y
z
jeff.bivin_at_lz95.org
17
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Now place the constants in the x column of
the numerator
X
x
y
z
jeff.bivin_at_lz95.org
18
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Now place the constants in the x column of
the numerator
X
x
y
z
jeff.bivin_at_lz95.org
19
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
X
x
y
z
jeff.bivin_at_lz95.org
20
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
x
y
z
Copy the remaining columns from the denominator
X
x
y
z
jeff.bivin_at_lz95.org
21
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
Copy the remaining columns from the denominator
X
jeff.bivin_at_lz95.org
22
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
Evaluate each determinant
X

jeff.bivin_at_lz95.org
23
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for x
X



jeff.bivin_at_lz95.org
24
now solve for y
jeff.bivin_at_lz95.org
25
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
Use determinants
y
jeff.bivin_at_lz95.org
26
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
27
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
28
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
1
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
29
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
1
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
30
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
31
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
32
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
33
4x 3y z -10x 12y 2z -5x 18y z
4
1
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
z
x
y
jeff.bivin_at_lz95.org
34
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
35
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
1
x
y
z
Fill the denominator's determinant with
the coefficients
y
x
y
z
jeff.bivin_at_lz95.org
36
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
x
y
z
Now place the constants in the y column of
the numerator
y
x
y
z
jeff.bivin_at_lz95.org
37
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
y
x
z
Now place the constants in the y column of
the numerator
y
x
y
z
jeff.bivin_at_lz95.org
38
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
y
x
z
Now place the constants in the y column of
the numerator
y
x
y
z
jeff.bivin_at_lz95.org
39
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
y
x
z
Now place the constants in the y column of
the numerator
y
x
y
z
jeff.bivin_at_lz95.org
40
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
y
x
z
Copy the remaining columns from the denominator
y
x
y
z
jeff.bivin_at_lz95.org
41
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
Copy the remaining columns from the denominator
y
jeff.bivin_at_lz95.org
42
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
Evaluate each determinant
y

jeff.bivin_at_lz95.org
43
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for y
y



jeff.bivin_at_lz95.org
44
now solve for z
jeff.bivin_at_lz95.org
45
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
Use determinants
z
jeff.bivin_at_lz95.org
46
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
47
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
48
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
1
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
49
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
1
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
50
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
51
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
52
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
53
4x 3y z -10x 12y 2z -5x 18y z
4
1
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
z
x
y
jeff.bivin_at_lz95.org
54
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
55
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
1
x
y
z
Fill the denominator's determinant with
the coefficients
z
x
y
z
jeff.bivin_at_lz95.org
56
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
x
y
z
Now place the constants in the z column of
the numerator
z
x
y
z
jeff.bivin_at_lz95.org
57
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
z
x
y
Now place the constants in the z column of
the numerator
z
x
y
z
jeff.bivin_at_lz95.org
58
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
z
x
y
Now place the constants in the z column of
the numerator
z
x
y
z
jeff.bivin_at_lz95.org
59
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
z
x
y
Now place the constants in the z column of
the numerator
z
x
y
z
jeff.bivin_at_lz95.org
60
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
z
x
y
Copy the remaining columns from the denominator
z
x
y
z
jeff.bivin_at_lz95.org
61
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
Copy the remaining columns from the denominator
z
jeff.bivin_at_lz95.org
62
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
Evaluate each determinant
z

jeff.bivin_at_lz95.org
63
4x 3y z -10x 12y 2z -5x 18y z
4
Solving for z
z



jeff.bivin_at_lz95.org
64
Final Result
4x 3y z -10x 12y 2z -5x 18y z
4
jeff.bivin_at_lz95.org