# Cramer Rule For 2 Equations

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.

## Cramer Rule for 2 Equations

 Rule 1 If ∆ ≠ 0. Then the system has unique solution and we can solve the equations by using the formula x = ∆ₓ/∆ , y = ∆ᵧ/∆ 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 non-zero then the system is inconsistent and it has no solution.

Example 1

Solve the  following equation using determinant method

x + 2y = 3

x + y = 2

Δ =

 1 2 1 1

=  1 - 2

= -1 ≠ 0              cramer rule for 2 equations

Δx =

 3 2 2 1

=  3 - 4

= -1 ≠ 0

Δy =

 1 3 1 2

=  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 =

 3 2 1 3

=  9 - 2

=  7 ≠ 0

Δx =

 5 2 4 3

=  15 - 8

=  7 ≠ 0

Δy =

 3 5 1 4

=  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 Cramer Rule for 2Unknowns to Minor of a Matrix 