4( -1,  y)T + 2( 3x, 10)T = (8, 24)T
 ( -4, 4y)T +  ( 6x, 20)T = (8, 24)T
 (6x-4, 4y+20)T           = (8, 24)T

6x-4  =  8
4y+20 = 24

6x = 12; x = 2
4y =  4; y = 1

Equating Corresponding Elements

In the original equation (and in the two re-writes of it) the "=" sign means "vector equality." Equating corresponding elements yields two equations where the "=" sign means "scalar equality" (or "real number equality.")

The "trick" of equating corresponding elements converts a column matrix equation into several scalar equations, one equation for each dimension of the column matrix. Then, sometimes, the scalar equations can be solved for the unknowns. But this does not always work.

Here is a case where it does not work: Find x and y :

4(x, 5)T + 2(y, 1)T = 2(12, 11)T 

This looks OK, but doing some work brings up a problem:

4( x,  5)T + 2( y, 1)T  =  2(12, 11)T 
 (4x, 20)T +  (2y, 2)T  =   (24, 22)T
 (4x+2y, 22 )T          =   (24, 22)T

4x + 2y  =  24
22       =  22

The two scalar equations we get by equating corresponding elements do not contain enough information to proceed.

a( -1, 5 )T + 2( 3x, 10 )T = ( 8, 25 )T