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The system Px + Qy = R has the solution (3,-1), where F, G, H, P, Q, and
Fx + Gy = H
R are nonzero real numbers.

Select all the systems that are also guaranteed to have the solution (3,-1)


a.) (P + F)x + (Q + G)Y = R + H

Fx + Gy = H

b.) (P + F)x + Qy = R + H

Fx + (G + Q)y = H

c.) Px + Qy = R

(3P + F)x + (3Q + G)y = 3H + R

d.) Px + Qy = R

(F - 2P)x +(G - 2Q)y = H - 2R

e.) Px + Qy = R

5Fx + 5Gy = 5H


*Please explain how you got your answer because I really don't understand this!

Sagot :

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

Step-by-step 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).

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