Answer
$$\left( {\bf{a}} \right)L\left( {x,y} \right) = 4x + y - 6,\,\,\,\left( {\bf{b}} \right)L\left( {2.1,2.99} \right) = 5.39$$
Work Step by Step
$$\eqalign{
& f\left( {x,y} \right) = xy + x - y;\,\,\,\,\left( {2,3} \right)\,\,\, \cr
& f\left( {2,3} \right) = \left( 2 \right)\left( 3 \right) + 2 - 3 \cr
& f\left( {2,3} \right) = 5 \cr
& {\text{The partial derivatives are}} \cr
& {f_x}\left( {x,y} \right) = y + 1 \cr
& {f_y}\left( {x,y} \right) = x - 1 \cr
& {\text{Evaluate the partial derivatives at }}\left( {2,3} \right) \cr
& {f_x}\left( {2,3} \right) = 3 + 1 = 4 \cr
& {f_y}\left( {2,3} \right) = 2 - 1 = 1 \cr
& \cr
& \left( a \right) \cr
& {\text{Therefore}}{\text{, the linear approximation to the function at }}\,\left( {2,3,5} \right){\text{ is}} \cr
& L\left( {x,y} \right) = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_y}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr
& L\left( {x,y} \right) = 4\left( {x - 2} \right) + \left( {y - 3} \right) + 5 \cr
& L\left( {x,y} \right) = 4x + y - 6 \cr
& \cr
& \left( b \right){\text{Estimate }}\,\left( {2.1,2.99} \right) \cr
& L\left( {2.1,2.99} \right) = 4\left( {2.1} \right) + 2.99 - 6 \cr
& L\left( {2.1,2.99} \right) = 5.39 \cr} $$
