Download 3-3 Cramer`s Rule

3-3 Cramer’s Rule
Cramer’s Rule is a process by which you can solve a system of equations. This lecture is
designed to show you how it works. If you want to see why it works, read pages 118 and
119. The text will give you a better understanding of why it works.
Cramer’s Rule makes use of something called determinants. A determinant is a square
array of numbers or variables. The quantities within a determinant are known as
elements.
a
b
c
d
This is a second order determinant. The value of this is ad-bc. To find the value of a
second order determinant, we find the difference between the two products of the
diagonal elements.
a
c
b
 ad  bc
d
Find the value of the determinant:
3
2
4
5
(3*5)-(2*4)=7
Multiply along the diagonals.
Simple enough, right?
Okay, so what does this have to do with a system of equations?
 Ax  By  C
Cramer’s Rule states the solution for a system of equations 
is (x,y);
 Dx  Ey  F
C B
A C
A B
F E
D F
0
where x 
, y
and
A B
A B
D E
D
E
D
E
So, what does all this mean? Look carefully at the system of equations. Note the capital
letters. These represent constants not variables. That is why I have capitalized them,
event though your text does not.
In order for Cramer’s Rule to be applied, note that the two equations MUST BE in
Standard Form.
If your equations are in standard form, it is then just a matter of plugging in values.
Solve the system using Cramer’s Rule:
6 x  7 y  9
x y 5
A=6 D=1
B=7 E=-1
C=-9 F=5
9 7
5
1
, y
x
6 7
1 1
6
1
6
1
(9 * 1)  (5 * 7)
(6 * 1)  (1 * 7)
9  35
x
67
 26
x
 13
x2
(6 * 5)  (1 * 9)
y
(6 * 1)  (1 * 7)
30  (9)
y
67
39
y
 13
y  3
(2,-3)
x
9
5
7
1
First, make sure that both equations are in
standard form. (They are). Next, write out
the constant values. Then, plug values in
to Cramer’s Rule.
Once you have written it in this format,
solve the determinants.
Solve for x.
Solve for y.
Now, you have the solution to your
equation.
That is how we use Cramer’s Rule to solve a system of equations. Seems complicated,
but really it is just a matter of extracting values and plugging them into a fomula.
3-4 – 3-7 Cannot be adequately posted due to the graphing that is necessary.