Solving a 3x3 system of equations using determinants

Solving a system of equations using determinants 3x3x system

a₁ x + b₁ y + c₁ z = d₁
a₂ x + b₂ y + c₂ z = d₂
a₃ x + b₃ y + c₃ z = d₃
variables are x , y & z.

This system of equations can be solved using substitution (or elimination). This is a tedious process!!! By observing the patterns in the expressions for x, y, and z one will notice similarities and symmetries. In fact, the solutions can once again be written in terms of appropriately defined determinants. Thus Cramer's Rule works in this case as well.

Specifically, define  ∆,  ∆_x,  ∆_y, ∆_z   as follows:

a₁ x + b₁ y + c₁ z = d₁
a₂ x + b₂ y + c₂ z = d₂
a₃ x + b₃ y + c₃ z = d₃
∆ =

a1	b1	c1
a2	b2	c2
a3	b3	c3

;

∆_x =

d1	b1	c1
d2	b2	c2
d3	b3	c3

;


∆_y =

a1	d1	c1
a2	d2	c2
a3	d3	c3

;


∆_z =

a1	b1	d1
a2	b2	d2
a3	b3	d3

We then find

x = ∆_x / ∆ ; y = ∆_y / ∆ ;  z = ∆_z / ∆
Cramer's Rule, 3x3


Example 1 - Solving a 3x3 system of equations using determinants