 ### Linear System Solver       © 2004, A2

If
 Ax +By = C Dx +Ey = F
then
 CE-BF x= AE-BD
and
 AF-CD y = AE-BD

 x=
 y =

1st: Arrange the equations so they look like:
 Ax +By = C Dx +Ey = F
2nd: Type in the values of constants and coefficients A, B, C, D, E, and F.
Enter negatives without a space -- enter "- 2" as "-2."
 x +y = x +y =
 For a "Quick Answer," press each:

3rd: Replace variables with thier values before evaluating the denominators of the solution.

AE - BD is
()() - ()().

4th: . AE - BD = .

5th: Determine the kind of solution the system has.
 numerator denominator fraction system solution(s), sample graph 0 0 0/0 is indeterminate dependent many solutions non 0 0 (non 0)/0 is undefined inconsistent no solution non 0 non 0 (non 0)/(non 0) = (solution) independent only 1 solution 6th: If the denominator is 0, this method won't produce a solution
either because there isn't a solution or because there are so many solutions.

If the denominator is not 0, the constant terms are required to compute the solution.
:   :
:   :
Write:

If
 Ax +By = C Dx +Ey = F
then
 CE-BF x= AE-BD
and
 AF-CD y = AE-BD

If
 Ax +By = C Dx +Ey = F
then
 x=
and
 y =

7th: Reduce as needed or, if needed, declare there to be no solution or multiple solutions.

How This Works

Using linear combination the equations for the lines are twice added together so that first one variable is eliminated and then the other. Each time an expression for the other variable is derived. These expressions may be used as formulas to eliminate the solving of the system algebraically and replace the solving with the simplication of a fraction.

The work is shown below.
 E(Ax +By = C) -B(Dx +Ey = F)

 AEx +BEy = CE -BDx -BEy = -BF (AE-BD)x +(BE-BE)y = (CF-BF) (AE-BD)x = (CF-BF)
and
 CE-BF x= AE-BD
 -D(Ax +By = C) A(Dx +Ey = F)

 -ADx -BDy = -CD ADx +AEy = AF (-AD+AD)x +(-BD+AE)y = (-CD+AF) (AE-BD)y = (AF-CD)
then
 AF-CD y = AE-BD

Cramer's Rule employs this computation in an easy format.

For more on Cramer's Rule, see cramers.xls, the spreadsheet.              © 2008, A. Azzolino www.mathnstuff.com/math/algebra/asyssol.htm