Question 190068
Find the value of x+y in the following

[3x-2] [x y] = [42 4]
..........[3 4]

Solution:
_________

we have a 1x2 matrix multiplied with a 2x2 matrix, which results in a 1x2 matrix again.

So, we first take the Left and side of this equation and multiply both the matrices and get the answers in terms of x and y.

we have:

[3x   -2] [x   y]
...............[3   4] 

=> {{{ [ ( (3x)(x) + (-2)(3) ) ..........  ( (3x)(y) + (-2)(4) ) ] }}}

we have got two terms which have been determined by multiplication, we now simplify them...

first element of the row is {{{ ( 3x^2 - 6 ) }}}
and, second element of the row is {{{ ( 3xy - 8 ) }}}

now we compare both sides of the original equation and we get:

{{{ 3x^2 - 6 = 42 }}} ..........(1)
and
{{{ 3xy - 8 = 4 }}} ............(2)

we have two equations and two variables x & y, we solve for them.

from equation (1) we have:

{{{ 3x^2 - 6 = 42 }}}
=> {{{ 3x^2 = 42 + 6 }}}
=> {{{ 3x^2 = 48 }}}
=> {{{ x^2 = 48/3 }}}
=> {{{ x^2 = 16 }}}
=> {{{ highlight(x = 4) }}} or {{{ highlight(x = -4) }}}

From equation (2),

when we plug in x = 4 , we get:

{{{ 3xy - 8 = 4 }}}
=> {{{ 3(4)y - 8 = 4 }}}
=> {{{ 12y  = 4 + 8 }}}
=> {{{ 12y = 12 }}}
=> {{{ y = 12/12 }}}
=> {{{ highlight(y=1) }}}

when we plug in x = -4 , we get:

{{{ 3xy - 8 = 4 }}}
=> {{{ 3(-4)y - 8 = 4 }}}
=> {{{ -12y  = 4 + 8 }}}
=> {{{ -12y = 12 }}}
=> {{{ y = 12/(-12) }}}
=> {{{ highlight(y=-1) }}}

Thus x+y = 4+1 = {{{ highlight(5) }}} or x+y = -4-1 = {{{ highlight(-5) }}}

Hope this helps.