# Thread: Linear combination - 2x2 Systems

1. ## Linear combination - 2x2 Systems

Hello.

1.-Let F be the field of complex numbers. Are the following two systems of linear equations equivalent?. If so, express each equation in each system as a linear combination of the equations in the other system.

$x-y= 0$
$2x+y= 0$

$3x+y= 0$
$x+y= 0$

Thanks a lot

2. Originally Posted by osodud
Hello.

1.-Let F be the field of complex numbers. Are the following two systems of linear equations equivalent?. If so, express each equation in each system as a linear combination of the equations in the other system.

$x-y= 0$
$2x+y= 0$

$3x+y= 0$
$x+y= 0$

Thanks a lot
I recommend that you answer the first question by trying to answer the second. If the two systems are equivalent then we can write:
x- y= a(3x+y)+ b(x+y) and 2x+ y= c(3x+y)+ d(x+y). Those are the same as
x- y= 3ax+ ay+ bx+ by and 2x+ y= 2cx+ cy or (1- 3a+ b)x- (1+ a+ b)y= 0 and (2- 3c- d)x+ (1- c- d)y= 0. In order that those equations be true for all x and y, we must have 1- 3a+ b= 0, 1+ a+ b= 0, 2- 3c- d= 0, and 1- c- d= 0. Are there a, b, c, d which satify those equations? If so, the two systems are equivalent.

If they are equivalent, to complete the solution you will need to determine i, j, p, and q such that 3x+ y= i(x-y)+ j(2x+y) and x+y= p(x-y)+ q(2x+y).