Question

If A= (7.00i -9.00j ) units, B = (-9.00i + 4.00j ) units, and C = (25.0i + 20.0j ) units, determine a and b such that a A + bB +C = 0.
a = ?
b = 2?

Answer 1

The condition aA + bB +C=0 gives a system of 2x2 linear system which you can solve by simple elimination. I'll give you the system. Simple algebra will do the rest...
7a - 9b = -25
-9a + 4b = -20

Answer 2

For condition aA+bB+C=0 can be written like
a(7.00i-9.00j)+b(-9.00i+4.00j)+(25.0i+20.0j)=0
(a7.00-b9.00+25.0)i +(-a9.00+b4.00+20.0)j =0
From the above equation if you campare the left hand side with right hand side you see that all the coefficient should be zero to satisfy the condition then
7a-9b+25=0
-9a+4b+20=0
Just solve the above two equation and you will get the answer