Solving a 2x2 system of equations using determinants

Solving a system of equations using determinants 2x2 system

a₁ x + b₁ y = c₁
a₂ x + b₂ y = c₂
variables are x & y.

If you solve this system of equations using substitution (or elimination) you will find:
x = (c₁ b₂  - b₁ c₂) / (a₁ b₂ - b₁ a₂) ; y = (a₁ c₂ - c₁ a₂) / (a₁ b₂ - b₁ a₂)

If we define 
∆ =

a1	b1
a2	b2

= a₁ b₂ - b₁ a₂


∆_x =

c1	b1
c2	b2

= c₁ b₂ - b₁ c₂




∆_y =

a1	c1
a2	c2

= a₁ c₂ - c₁ a₂

It follows that:
Cramer's Rule 2x2

x = ∆_x / ∆ =

c1	b1
c2	b2

/

a1	b1
a2	b2

y = ∆_y / ∆ =

a1	c1
a2	c2

/

a1	b1
a2	b2

Thus, the solution of the system can be expressed using determinant ratios. This result is known as Cramer's Rule.

Note:
a₁ x + b₁ y = c₁
a₂ x + b₂ y = c₂

Can find x in system above by eliminating y...
b₂ . (a₁ x + b₁ y = c₁)
b₁ . (a₂ x + b₂ y = c₂)

(1) ==> a₁ b₂ x + b₁ b₂ y = c₁ b₂
(2) ==> a₂ b₁ x + b₂ b₁ y = c₂ b₁

(1) - (2) : a₁ b₂ x - b₁ a₂ x = c₁ b₂ - b₁ c₂
\ (a₁ b₂ - b₁ a₂) x = c₁ b₂ - b₁ c₂
\ x = (c₁ b₂ - b₁ c₂) / (a₁ b₂ - b₁ a₂)

x = ∆_x / ∆ =

c1	b1
c2	b2

/

a1	b1
a2	b2

Similarly you could find y by eliminating x in system of equations.
If this is done, you will find...
y = (a₁ c₂ - c₁ a₂) / (a₁ b₂ - b₁ a₂)
y = ∆_y / ∆ =

a1	c1
a2	c2

/

a1	b1
a2	b2

Example 1 - Use Cramer's Rule to express the solution of a system in Determinant Form
Example 2 - Use Cramer's Rule to express the solution of a system in Determinant Form