Cramer's Rule
Gabriel Cramer was a Swiss mathematician (1704-
1752)
Reference from:
:
Fundamentals Methods of Mathematical
Economics
4
th
Edition (Page 103-107)

Introduction
Cramer’s Rule is a method for solving linear
simultaneous equations. It makes use of
determinants and so a knowledge of these is
necessary before proceeding.
Cramer’s Rule relies on determinants

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
cx+dy=f
d
c
b
a

Cramer’s Rule for 2x2 System
Let A be the coefficient matrix
Linear System
Coeff Matrix
ax+by=e
cx+dy=f
If detA
0, then the system has exactly one
solution:
and
= ad – bc
A
d
f
b
e
x
det
A
f
c
e
a
y
det
d
c
b
a

Key Points
The denominator consists of the coefficients
of variables (x in the first column, and y in the
second column).
The numerator is the same as the
denominator, with the constants replacing the
coefficients of the variable for which you are
solving.

Example -
Applying Cramer’s Rule
on a System of Two Equations
Solve the system:
8x+5y= 2
2x-4y= -10
The coefficient matrix is:
and
So:
and
4
2
5
8
42
)
10
(
)
32
(
4
2
5
8
42
4
10
5
2
x
42
10
2
2
8
y

Solution: (-1,2)
1
42
42
42
)
50
(
8
42
4
10
5
2

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