Let the length of the garden be x and the width be y.
It is given that the width of a rectangular garden is 11 feet longer than 3 times its length so it follows:
[tex]\begin{gathered} y=11+3x \\ 3x-y=-11\ldots(i) \end{gathered}[/tex]It is also given that the garden's perimeter is 150 feet so it follows:
[tex]\begin{gathered} 2(x+y)=150 \\ x+y=75\ldots(ii) \end{gathered}[/tex]Add equation (i) and (ii) to get:
[tex]\begin{gathered} 3x-y=-11 \\ + \\ x+y=75 \\ 4x=64 \\ x=16 \end{gathered}[/tex]Substitute x=16 in (ii) to get:
[tex]\begin{gathered} 16+y=75 \\ y=59 \end{gathered}[/tex]Therefore the dimensions are:
Length=16 feet
Width=59 feet.