Cramer's Rule and a System of Two Equations

Date: 12/27/2003 at 17:40:43
From: Brittany
Subject: using Cramer's Rule

Find x and y using Cramer's Rule:

 2x - 6y = 12
-4x + 3y = 7

Date: 12/27/2003 at 19:38:01
From: Doctor Rob
Subject: Re: using Cramer's Rule

Thanks for writing to Ask Dr. Math, Brittany!

Cramer's Rule says that in such situations, x and y can be written as
the quotient of two determinants.  The denominator is the same for
both x and y:  it's two-by-two and contains the coefficients of x and
y from the two equations as entries, one row for each equation and one
column for each variable x and y:

 2x - 6y  -->  | 2 -6|
-4x + 3y  -->  |-4  3|

The numerator for x is found by replacing the column containing the
coefficients of x by a column containing the constants from the right
side of the equations:

 2x - 6y = 12  -->  |12 -6|
-4x + 3y = 7   -->  | 7  3|

The numerator for y is found by replacing the column containing the
coefficients of y by a column containing the constants from the right
side of the equations:

 2x - 6y = 12  -->  | 2 12|
-4x + 3y = 7   -->  |-4  7|

Compute these determinant values, and then take the quotients
indicated above, and you'll have the solution values of x and y:

        |12 -6|
        | 7  3|    (12)(3) - (7)(-6)   36 - (-42)    78    13
   x = --------- = ----------------- = ---------- = ---- = --
        | 2 -6|    (2)(3) - (-4)(-6)    6 - 24      -18    -3
        |-4  3|

        | 2 12|
        |-4  7|
   y = --------- = ?
        | 2 -6|
        |-4  3|
        
Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/