The sum of two numbers is zero. When - 2 times the smaller number is added to 2 times the larger, the result is 3. What's the two numbers?

Smaller number = X

Larger number = Y

Equation 1: X + Y = 0

Equation 2: (-2 * X) + (2*Y) = 3

Solve for Y in Equation 2:

(-2 * X) + (2*Y) = 3

Add 2X to each side:

2 * y = 2*X + 3

Divide each term by 2:

y = X + 3/2

Replace Y in Equation 1 with the above solution for y:

X + (X + 3/2) = 0

Simplify:

2X + 3/2 = 0

Subtract 3/2 from each side"

2X = - 3/2

Divide both sides by 2:

X = - 3/2 / 2

X = - 3/4

Now replace X in the fist equation to solve for Y:

-3/4 + Y = 0

y = 3/4

The two numbers are - 3/4 and 3/4