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.