The perimeter of the given rectangle is 78 units.
Recall that the perimeter of a rectangle is given by
[tex]P=2(w+l)[/tex]Where w is the width and l is the length of the rectangle.
As you can see from the given figure,
w = 3z + 3
l = 4z + 1
We are asked to find the side length of side PS.
Substitute the given values into the above formula and solve for z.
[tex]\begin{gathered} P=2(w+l) \\ 78=2(3z+3+4z+1_{}) \\ 78=2(7z+4_{}) \\ \frac{78}{2}=(7z+4_{}) \\ 39=7z+4_{} \\ 39-4=7z \\ 35=7z \\ \frac{35}{7}=z \\ 5=z \end{gathered}[/tex]So, the value of z is 5
Finally, the length of side PS is
[tex]\begin{gathered} PS=4z+1 \\ PS=4(5)+1 \\ PS=20+1 \\ PS=21 \end{gathered}[/tex]Therefore, the length of the side PS is 21 units.