Cramer’s Rule for 2x2 systems
Jerry, I'm at the corner
of 1st and 1st.
Wait a minute, how can
a street intersect itself?
I must be at the nexus
of the universe!
Cramer’s Rule
Actually, Gabriel Cramer was a
Swiss mathematician (1704 -1752)
Coefficient Matrices



You can use determinants to solve a system
of linear equations.
You use the coefficient matrix of the linear
system.
Linear System
Coeff Matrix
ax+by=e
a b 


cx+dy=f
c d


Cramer’s Rule for 2x2 System


Let A be the coefficient matrix
Linear System
Coeff Matrix
ax+by=e
a b 


cx+dy=f
c d


If detA
x


0, then the system has exactly one solution:
e
b
a
e
f
d
c
f
det A
and
y 
det A
Example 1- Cramer’s Rule 2x2



Solve the system:
8x+5y=2
2x-4y=-10
The coefficient matrix is:  8

2
So:
x
2
5
 10
4
 42
5 

 4
and
8
5
2
4
and
y 
 (  32 )  (10 )   42
8
2
2
 10
 42
x
y 
2
5
 10
4
 42
8
2
2
 10
 42


 8  (  50 )
 42
 80  4
 42
Solution: (-1,2)


42
 42
 84
 42
2
 1
Example 2- Cramer’s Rule 2x2



Solve the system:
2x+y=1
3x-2y=-23
The solution is: (-3,7) !!!
Example 3- Cramer’s Rule 2x2



Solve the system:
5x+2y=8
2x-3y= 7
The solution is: (2,-1) !!!