Question

How do you solve `((4, 8), (2, 5))((x), (y))=((0),(6))`?

Answer 1

`((x),(y))=((-12),(6))`

Explanation:

To solve `((4,8),(2,5))((x),(y))=((0),(6))`, we need to first find inverse of the matrix `((4,8),(2,5))`.

The general form for inverse of matris `((a,b),(c,d))` is `1/(ad-bc)((d,-b),(-c,a))`.

Hence inverse of `((4,8),(2,5))` is `1/(4*5-8*2)((5,-8),(-2,4))` or `1/4((5,-8),(-2,4))`. Multiplying both sides of `((4,8),(2,5))((x),(y))=((0),(6))` by this gives

`((x),(y))=1/4((5,-8),(-2,4))((0),(6))` .............(A)

as `1/4((5,-8),(-2,4))((4,8),(2,5))=((1,0),(0,1))`

Solving (A) by simple matrix multiplication gives us

`((x),(y))=1/4((-48),(24))=((-12),(6))`

Answer 2

`(x,y)=(-12,6)`

Explanation:

(this is just an alternative method to that provided by Shwetank Mauria that doesn't require the use of an inverse matrix)

`((4,8),(2,5))xx((x),(y)) = ((4x+8y),(2x+5y))` (by standard matrix multiplication)

Therefore
`color(white)("XXX")((4,8),(2,5))xx((x),(y))= ((0),(6))`
is equivalent to
[1]`color(white)("XXX")4x+8y=0`
[2]`color(white)("XXX")2x+5y=6`

If we multiply equation [2] by `2` and subtract the result form equation [1], we get
[3]`color(white)("XXX")-2y=-12`
and from this
[4]`color(white)("XXX")y=6`

Substituting `6` for `y` back in equation [1] gives
[5]`color(white)("XXX")cancel(4)x+cancel(8)^2xx(6)=0`
and
[6]`color(white)("XXX")x=-12`