If A and B are non-colinear vectors, how can we solve this?

Question

and P = (2x + 3y - 2)A + (3x + 2y + 5)B
Q = (-x + 4y - 2)A + (3x - 4y + 7)B

find x, y; such that 7P = 3Q.

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Is it so that P & Q are colinear vectors as 7P = 3Q and we need to do something of it? How?

Please help.

Answer 1

7P = (14x + 21y - 14)A + (21x + 14y + 35)B
3Q = (-3x + 12y - 6)A + (9x - 12y + 21)B
As they are equal, then the coeficients of each A and B are equal:
14x+21y-14 = 21x+14y+35 and
3x+12y-6 = 9x-12y+21
which gives you two equations in two unknowns. The rest is easy.

Answer 2

What kind of vectors are A,B? Is this all in R^2?

Write out 7P-3Q = 0 in matrix form where A,B are elements of a column vector:

Mv=0, where M comes from the equation and v = (A,B)^t (transpose)

or

Mv=0, where v=(x,y)^t....

see what the system looks like.