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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
 
 
64863.png ...(2)
where 64857.png.
 
Solving by cross-multiplication, we have
64851.png
or 64845.png

System of linear equations in three variables

Let the given system of linear equations in three variables x, y, and z be
 
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2  ...(3)
a3x + b3y + c3z = d3
 
Let
Δ = 64839.png, Δ1 = 64833.png,
Δ2 = 64827.png, Δ3 = 64821.png

Solutions under different conditions

  1. If Δ 0, then given system of equations is consistent and it has unique (one) solution which is given by
    x = 64815.png, y = 64808.png, and z = 64802.png
  2. If Δ = 0 and at least one of Δ1, Δ2, and Δ3 is non-zero, then given system of equations is inconsistent and it will have no solution.
  3. If all of Δ, Δ1, Δ2, and Δ3 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
 
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
a3x + b3y + c3 = 0
 
will be consistent if the values of x and y obtained from any two equations satisfy the third equation,
 
or 64796.png
 
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.




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