Answer
$y=-6$
Work Step by Step
Using $y-y_1=\dfrac{y_1-y_2}{x_1-x_2}(x-x_1)$ or the two-point form of linear equations, the equation of the line passing through $\left(
-\dfrac{4}{9},-6
\right)$ and $\left(
\dfrac{12}{7},-6
\right)$ is
\begin{array}{l}\require{cancel}
y-(-6)=\dfrac{-6-(-6)}{-\frac{4}{9}-\frac{12}{7}}\left( x-\left(-\dfrac{4}{9}\right) \right)
\\\\
y+6=\dfrac{-6+6}{-\frac{4}{9}-\frac{12}{7}}\left( x+\dfrac{4}{9}\right)
\\\\
y+6=\dfrac{0}{-\frac{4}{9}-\frac{12}{7}}\left( x+\dfrac{4}{9}\right)
\\\\
y+6=0
\\\\
y=-6
.\end{array}
