How do you solve # x+2y=9# and #x+y=1# using matrices?

1 Answer
Feb 27, 2016

Answer:

One method is to use Cramer's Rule (with determinants)

Explanation:

The given equations can be written as a matrix:
#color(white)("XXXX")xcolor(white)("X")ycolor(white)("X")=c#
#color(white)("XXX")((color(red)(1),color(blue)(2),,color(green)(9)),(color(red)(1),color(blue)(1),,color(green)(1)))#

If #D# is the determinant formed by the #x# and #y# columns of this matrix:
#color(white)("XXX")D=|(color(red)(1),color(blue)(2)),(color(red)(1),color(blue)(1))| = color(red)(1)xxcolor(blue)(1) - color(red)(1)xxcolor(blue)(2) = -1#

and #D_x# is the determinant of the same submatrix but with the #x# column replaced with the #c# column
#color(white)("XXX")D_x=|(color(green)(9),color(blue)(2)),(color(green)(1),color(blue)(1))|=color(green)(9)xxcolor(blue)(1)-color(green)(1)xxcolor(blue)(2)=7#

and #D_y# is the determinant fo the same submatrix as #D# but with the #y# column replaced with the #c# column
#color(white)("XXX")D_y=|(color(red)(1),color(green)(9)),(color(red)(1),color(green)(1))|=color(red)(1)xxcolor(green)(1)-color(red)(1)xxcolor(green)(9)=-8#

Cramer's Rule tells us
#color(white)("XXX")x= (D_x)/(D) = 7/(-1) = -7#
and
#color(white)("XXX")y=(D_y)/(D) = (-8)/(-1) = 8#