do the Two systems of equations shown have the same solution?

Question

4x+y=−1
x+y=2

8x+2y=−2
−4x+4y=−8

Answer 1

Yes, the two systems of equations have the same solution.

For the first system of equations:
4x+y=−1
x+y=2

If we subtract the second equation from the first equation, we get:
4x+y - (x+y) = -1 - 2
3x = -3
x = -1
Substituting x=-1 back into the second equation:
-1 + y = 2
y = 3

So the solution is x = -1, y = 3.

For the second system of equations:
8x+2y=−2
−4x+4y=−8

If we divide the first equation by 2, we get:
4x + y = -1

Subtracting the second equation from the first equation, we have:
(4x + y) - (-4x + 4y) = -1 - (-8)
8x - 3y = 7
Rearranging this equation:
8x = 3y + 7

If we compare this equation with the first equation in the first system of equations, we can see that they are equivalent. Therefore, the second system of equations also has the solution x = -1, y = 3, which is the same as the solution for the first system of equations.