Given:
The vector (-3,2) describes the translations:
[tex]\begin{gathered} A(-1,x)\rightarrow A^{\prime}(-4y,1) \\ B(2z-1,1)\rightarrow B^{\prime}(3,3) \end{gathered}[/tex]From the first translation, we will find the values of (x) and (y)
So, we have the rule:
[tex](-1,x)+(-3,2)=(-4y,1)[/tex]The sum of the x-coordinates from the left will equal the x-coordinates from the right
[tex]-1-3=-4y[/tex]Solve the equation to find y
[tex]\begin{gathered} -4=-4y\rightarrow(\div-4) \\ y=1 \end{gathered}[/tex]Now, The sum of the y-coordinates from the left will equal the y-coordinates from the right
[tex]x+2=1[/tex]Subtract 2 from both sides:
[tex]x=-1[/tex]From the second translation, we will find the value of z:
[tex]\begin{gathered} (2z-1,1)+(-3,2)=(3,3) \\ 2z-1-3=3 \\ 2z-4=3 \\ 2z=3+4 \\ 2z=7 \\ z=\frac{7}{2}=3.5 \end{gathered}[/tex]so, the answer will be:
[tex]\begin{gathered} x=-1 \\ y=1 \\ z=3.5 \end{gathered}[/tex]See the following figure:
