Answer to Question #91451 in Algebra for Ra

Question

Consider the equation E ≡ 5x-2y=3.
Write down equations E₁, E₂, E₃, respectively so that
1.) E and E₁ are inconsistent;
2.) E and E₂ have a unique solution;
3.) E and E₃ have infinitely many solutions.

Answer 1

1) E1: 5x-2y=5

Proof:

Consider the system of equations E and E1. Subtracting the second equation from the first equation we'll get 0 = 2, that is incorrect.

2) E2: 2x-5y=0

Proof:

Let's solve the system of equations E and E2. From the E2 we can express x=5y/2. Then

5(5y/2)-2y=3;

25y/2-2y=3;

21y/2=3

21y=6;

y=2/7;

x=5y/2=5(2/7)/2=5/7

3) E3:10x-4y=6

Solution: To get a system of equations with infinitely many solutions let's multiply the equation E by a constant (in our case 2) to make them dependent.