What are the two numbers if one half of the sum is negative three and one half of the difference is seven?

To find the two numbers based on the conditions provided in the problem, we can set up a system of equations. Let's denote the two numbers as (x) and (y).

The problem states that half of the sum is negative three:

[

\frac{x + y}{2} = -3

]

Multiplying both sides by 2 gives:

[

x + y = -6 \quad \text{(1)}

]

It also mentions that half of the difference is seven:

[

\frac{x - y}{2} = 7

]

Multiplying both sides by 2 gives:

[

x - y = 14 \quad \text{(2)}

]

Now, we have a system of two linear equations:

1. (x + y = -6)

2. (x - y = 14)

To solve for (x) and (y), we can add the two equations:

[

(x + y) + (x - y) = -6 + 14

]

This simplifies to:

[

2x = 8

]

From here, we can solve for (x):

[

x =