Question

How do you solve `x-2y=2` and `3x-5y=9` using matrices?

Answer 1

`x=8, y=3`

Explanation:

Put the eqns in matrix form

`x-2y=2`
`3x-5y=9`

become

`((1,-2),(3,-5))` `((x),(y))=((2),(9))`

we therefore have

`M((x),(y))=((2),(9))`

by premultiplying both sides by `M^(-1)`

`M^(-1)M((x),(y))=M^(-1)((2),(9))`

`I((x),(y))=M^(-1)((2),(9))`

where `I=((1,0),(0,1))`

`:.((x),(y))=M^(-1)((2),(9))`

so we need to find the inverse of the matrix.

`M=((1,-2),(3,-5))`

for any `2xx2` mx

`M=((a,b),(c,d))`

`M^(-1)=1/Delta((d,-b),(-c,a))" providing "DeltaM!=0`

find its determinant

`Delta M=|(1,-2),(3,-5)|`

`DeltaM=1xx-5-(3xx-2)=-5--6=1`

`because Delta M!=0 " the inverse exists"`

`:.M^(-1)=1/1((-5,2),(-3,1))=((-5,2),(-3,1))`

so`((x),(y))=M^(-1)((2),(9))=((-5,2),(-3,1))((2),(9))`

`((x),(y))=((-5xx2+2xx9),(-3xx2+1xx9))`

`((x),(y))=((-10+18),(-6+9))`

`((x),(y))=((-10+18),(-6+9))`

`((x),(y))=((8),(3))`