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Cramer's Rule (Three Equations)

cramer3.html

Exploration

Show that the solution to the system of three linear equations in three unknowns

"cramer3_1.gif"

is given by

"cramer3_2.gif", "cramer3_3.gif" and "cramer3_4.gif"

where

"cramer3_5.gif".

Approach

Use the Mathematica Det command to compute the appropriate determinants and then substitute the solutions back into the original equations to demonstrate that they solve the equations.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Compute the necessary determinants.

Clear[a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3];
dx=Det[{{-d1,b1,c1},{-d2,b2,c2},{-d3,b3,c3}}];
dy=Det[{{a1,-d1,c1},{a2,-d2,c2},{a3,-d3,c3}}];
dz=Det[{{a1,b1,-d1},{a2,b2,-d2},{a3,b3,-d3}}];
dD=Det[{{a1,b1,c1},{a2,b2,c2},{a3,b3,c3}}];

Compute the solutions.

{x1,y1,z1}={dx/dD,dy/dD,dz/dD}//Simplify

"cramer3_6.gif"

Show that the solutions solve the original equations.

Clear[x,y];
{a1*x+b1*y+c1*z+d1,
a2*x+b2*y+c2*z+d2,
a3*x+b3*y+c3*z+d3} /.
   {x->x1,y->y1,z->z1} //Simplify

"cramer3_7.gif"

Discussion

The Solve command produces the same result.

Clear[z];
Simplify[
   Solve[{a1*x+b1*y+c1*z+d1==0,
          a2*x+b2*y+c2*z+d2==0,
          a3*x+b3*y+c3*z+d3==0},{x,y,z}]
   ]

"cramer3_8.gif"


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