To solve the system of equations given by:
- \( y = -\frac{1}{5}x + 2 \)
- \( y = \frac{1}{5}x + 4 \)
we can set the equations equal to each other since they both equal \( y \):
\[ -\frac{1}{5}x + 2 = \frac{1}{5}x + 4 \]
To eliminate the fractions, we can multiply through by 5:
\[ -1x + 10 = 1x + 20 \]
Now, we'll combine like terms:
\[ 10 - 20 = 1x + 1x \] \[ -10 = 2x \]
Now, divide both sides by 2:
\[ x = -5 \]
Now that we have the value of \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use the first equation:
\[ y = -\frac{1}{5}(-5) + 2 \] \[ y = 1 + 2 \] \[ y = 3 \]
Thus, the solution to the system of equations is:
\[ \boxed{(-5, 3)} \]
This means that the two lines intersect at the point (-5, 3).