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 = ∆/∆ , y = ∆/∆              Examples


Rule 2

If ∆ = 0 and ∆ₓ= 0, ∆ᵧ= 0  and at least one of the coefficients a₁,a₂,a₁,a₂ is non zero,then the system is consistent and has infinitely many solution.   

Examples


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.           Examples

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
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