To find the y-coordinate of the solution to the system of linear equations, we need to solve the equations:
- \( y = -3x + 2 \)
- \( 5x + 3y = 0 \)
We can substitute the expression for \( y \) from the first equation into the second equation.
Substituting \( y \) into the second equation gives:
\[ 5x + 3(-3x + 2) = 0 \]
Simplifying this:
\[ 5x - 9x + 6 = 0 \] \[ -4x + 6 = 0 \]
Now, solve for \( x \):
\[ -4x = -6 \] \[ x = \frac{6}{4} = \frac{3}{2} \]
Now that we have the value of \( x \), we can find the corresponding \( y \) coordinate by substituting \( x = \frac{3}{2} \) back into the first equation:
\[ y = -3\left(\frac{3}{2}\right) + 2 \] \[ y = -\frac{9}{2} + 2 \] \[ y = -\frac{9}{2} + \frac{4}{2} \] \[ y = -\frac{5}{2} \]
Thus, the y-coordinate of the solution to this system is
\[ \boxed{-\frac{5}{2}} \]