Let (x1, y1) and (x2, y2) both be solutions to the 2 times 2 system a11x + a12y

Question

Let (x1, y1) and (x2, y2) both be solutions to the 2 times 2 system a11x + a12y = 0 a21x + a22y = 0 Show that for any real number beta, the ordered pair (betax1, betay1) is also a solution. Show that (x1, y1) +(x2, y2) is also a solution. [Hint: Rewrite the system as a matrix equation, where A has entries aij.]

Explanation / Answer

(x1,y1), (x2,y2) are solutions.. => a11 x1+ a12 y1=0, a21 x1+ a22 y1 =0 (eqn 1,2) a11 x2+ a12 y2=0, a21 x2+ a22 y =0 (eqn 3,4) Now, multiply equation 1 and 2 by B (or beta), we have a11 Bx1+ a12 By1=0 a21 Bx1+ a22 By1 =0 So, (Bx1,By1) also satisifes the set of equations.. so it's a solution... b) Add eqn 1,3 and 2,4 a11(x1+x2)+ a12 (y1+y2)=0 a21(x1+x2)+ a22 (y1+y2)=0 =>(x1+x2,y1+y2) is also a solution of the set of equations..

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