How do you solve #6x + 2y = -16# and #5x + 6y = 4# using matrices?

1 Answer
David B.
Mar 10, 2016

Use the matrix equation and the inverse matrix to find #[(x),(y)]= [(-4),(4)]#

Explanation:

We begin by writing our system of equations in matrix form

#[(6,2), (5,6)][(x),(y)]=[(-16),(4)]#

Let #A=[(6,2), (5,6)]#

We then use the identity that a matrix, #A# multiplied by its inverse, #A^(-1)# is the identity matrix, #I#, i.e.

#A^(-1)A=A A^(-1)=I#

Multiplying both sides of our original matrix equation by the inverse matrix we get:

#A^(-1)A[(x),(y)]=A^(-1)[(-16),(4)]#

which simplifies to

#[(x),(y)]=A^(-1)[(-16),(4)]#

since any matrix or vector multiplied by the identity matrix is itself. We now need to find #A^(-1)#. Use the method described here How do I find the inverse of a #2xx2# matrix? to find the inverse to be:

#A^(-1)=[(3/13,-1/13), (-5/26,3/13)]#

Substitute this into the equation to find #x# and #y#

#[(x),(y)]=[(3/13,-1/13), (-5/26,3/13)][(-16),(4)] = [(-4),(4)]#

Therefore #x=-4# and #y=4#

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