Answer:
The systems that are guaranteed have the solution (3, -1) are;
a.) (P + F)·x + (Q + G)·y = R + H
F·x + G·y = H
d.) P·x + Q·y = R
(F - 2·P)·x + (G - 2·Q)·y = H - 2·R
e) P·x + Q·y = R
5·F·x + 5·G·y = 5·H
Explanation:
The given system of equation are;
a) P·x + Q·y = R...(1)
F·x + G·y = H...(2)
The solution of the system of equation = (3, -1)
Adding equation (1) to equation (2) gives;
P·x + Q·y + F·x + G·y = R + H
(P + F)·x + (Q + G)·y = R + H...(3)
Therefore, given that equation (3) is obtained from equation (1) and (2) by addition, equation (3), (P + F)·x + (Q + G)·y = R + H, we have;
The system of equation;
(P + F)·x + (Q + G)·y = R + H
F·x + G·y = H, derived from the given system of equation Is bound to have the same same solution (3, -1) as the given system of equation
d.) By multiplying equation (1) by 2, we have;
2 × (P·x + Q·y) = 2 × R
2·P·x + 2·Q·y = 2·R...(4)
Subtracting equation (4) from equation (2) gives;
F·x + G·y - (2·P·x + 2·Q·y) = H - 2·R
F·x - 2·P·x + G·y - 2·Q·y = H - 2·R
(F - 2·P)·x + (G - 2·Q)·y = H - 2·R
Therefore, for the following system, obtained from the original system, we have that the solution is (3, -1);
P·x + Q·y = R
(F - 2·P)·x + (G - 2·Q)·y = H - 2·R
e) For the system of equation, we have;
P·x + Q·y = R
5·F·x + 5·G·y = 5·H
The above system of equation is obtained from the original system by multiplying equation (2) by 5, therefore the solution of the system P·x + Q·y = R, 5·F·x + 5·G·y = 5·H is the same as the solution for the original system of equations (3, -1).