In this page cramer rule for 2 equations we are going to see procedure and example problems of solving 2 unknowns using cramer rule. The another name of cramer rule method is determinant method.
Rule 1 
If ∆ ≠ 0. Then the system has unique solution and we can solve the equations by using the formula x = ∆ₓ/∆ , y = ∆ᵧ/∆ Examples 
Rule 2 
If ∆ = 0 and ∆ₓ= 0, ∆ᵧ= 0 and at least one of the coefficients a₁₁,a₁₂,a₂₁,a₂₂ is non zero,then the system is consistent and has infinitely many solution. 
Rule 3 
If ∆ = 0 and at least one of the values ∆ₓ, ∆ᵧ is nonzero then the system is inconsistent and it has no solution. Examples 
Example 1
Solve the following equation using determinant method
x + 2y = 3
x + y = 2
Δ = 

= 1  2
= 1 ≠ 0 cramer rule for 2 equations
Δ_{x} = 

= 3  4
= 1 ≠ 0
Δ_{y} = 

= 2  3
= 1 ≠ 0
Here ∆ ≠ 0, ∆x ≠ 0 and ∆y ≠ 0. Then the system is consistent and it has unique solution. By cramer's rule.
x = ∆x/∆ x = (1)/(1) x = 1 
y = ∆y/∆ y = (1)/(1) y = 1 
Solution:
x = 1
y = 1
Example 2
Solve the following equation using determinant method
3x + 2y = 5
x + 3y = 4
Δ = 

= 9  2
= 7 ≠ 0
Δ_{x} = 

= 15  8
= 7 ≠ 0
Δ_{y} = 

= 12  5
= 7 ≠ 0
Here ∆ ≠ 0, ∆x ≠ 0 and ∆y ≠ 0. Then the system is consistent and it has unique solution. By cramer's rule.
x = ∆x/∆ x = 7/7 x = 1 
y = ∆y/∆ y = 7/7 y = 1 
Solution:
x = 1
y = 1