Answer
The equation of the tangent plane at $P = \left( {0,3, - 1} \right)$ is $4x + y - 2z = 5$.
Work Step by Step
Write the surface:
$F\left( {x,y,z} \right) = xy + y - z{{\rm{e}}^x} - {{\rm{e}}^{z + 1}} = 3$
The partial derivatives are
${F_x} = y - z{{\rm{e}}^x}$, ${\ \ \ }$ ${F_y} = x + 1$, ${\ \ \ }$ ${F_z} = - {{\rm{e}}^x} - {{\rm{e}}^{z + 1}}$
At $P = \left( {0,3, - 1} \right)$,
${F_x}\left( {0,3, - 1} \right) = 4$, ${\ \ \ }$ ${F_y}\left( {0,3, - 1} \right) = 1$, ${\ \ \ }$ ${F_z}\left( {0,3, - 1} \right) = - 2$
By Theorem 5 of Section 15.5, the tangent plane to the surface at $P = \left( {0,3, - 1} \right)$ has equation
${F_x}\left( {0,3, - 1} \right)\left( {x - 0} \right) + {F_y}\left( {0,3, - 1} \right)\left( {y - 3} \right) + {F_z}\left( {0,3, - 1} \right)\left( {z + 1} \right) = 0$
$4x + \left( {y - 3} \right) - 2\left( {z + 1} \right) = 0$
$4x + y - 2z = 5$
So, the equation of the tangent plane at $P = \left( {0,3, - 1} \right)$ is $4x + y - 2z = 5$.
