Let's denote the smaller number as \(x\) and the larger number as \(y\).
From the given information, we have two equations:
1. \(x + y = 0\) (The sum of the numbers is zero)
2. \(5x + 3y = 1\) (5 times the smaller number added to 3 times the larger number equals 1)
We can solve this system of equations using substitution or elimination method. Let's use the elimination method here.
From the first equation, we can express \(y\) in terms of \(x\):
\(y = -x\)
Substituting this expression for \(y\) into the second equation:
\(5x + 3(-x) = 1\)
Simplifying:
\(5x - 3x = 1\)
\(2x = 1\)
\(x = \frac{1}{2}\)
Now that we know the value of \(x\), we can find \(y\) using the first equation:
\(x + y = 0\)
\(\frac{1}{2} + y = 0\)
\(y = -\frac{1}{2}\)
So, the two numbers are \(\frac{1}{2}\) and \(-\frac{1}{2}\).