# Cramer*'*s Rule

Use of determinants in solving linear equations with the help of Cramer

*'*s rule:# System of linear equations in two variables

Let the given system of equations be

...(2)

where .

Solving by cross-multiplication, we have

or

# System of linear equations in three variables

Let the given system of linear equations in three variables

*x*,*y*, and*z*be*a*

_{1}

*x*+

*b*

_{1}

*y*

*+*

*c*

_{1}

*z*=

*d*

_{1}

*a*

_{2}

*x*+

*b*

_{2}

*y*+

*c*

_{2}

*z*=

*d*

_{2 ...}(3)

*a*

_{3}

*x*+

*b*

_{3}

*y*+

*c*

_{3}

*z*=

*d*

_{3}

Let

Î” = , Î”

_{1}= ,Î”

_{2 }= , Î”_{3 }=# Solutions under different conditions

- If Î” â‰ 0, then given system of equations is consistent and it has unique (one) solution which is given by
*x*= ,*y*= , and*z*= - If Î” = 0 and at least one of Î”
_{1}, Î”_{2}, and Î”_{3}is non-zero, then given system of equations is inconsistent and it will have no solution. - If all of Î”, Î”
_{1}, Î”_{2}, and Î”_{3}are zero, then given system of equations is consistent and has infinitely many solutions.

# Conditions for consistency of three linear equations in two unknowns

System of three linear equations in

*x*and*y**a*

_{1}

*x*+

*b*

_{1}

*y*+

*c*

_{1}= 0

*a*

_{2}

*x*+

*b*

_{2}

*y*+

*c*

_{2}= 0

*a*

_{3}

*x*+

*b*

_{3}

*y*+

*c*

_{3}= 0

will be consistent if the values of

*x*and*y*obtained from any two equations satisfy the third equation,or

This is the required condition for consistency of three linear equations in two unknowns. If such system of equations is consistent, then the number of solutions is one.