Question

Convert the equalities below equalities by adding sack variables. Use \(\displaystyle{s}_{{1}}\ \text{ and }\ {s}_{{2}}\) for your slack variables.
\(\displaystyle{3}{x}_{{1}}+{9}{x}_{{2}}\leq{42}\) converts to () =42
\(\displaystyle{15}{x}_{{1}}+{7}{x}_{{2}}\leq{38}\) converts to () = 38

Answer 1

Step 1
Solution: Given:
\(\displaystyle{3}{x}_{{1}}+{9}{x}_{{2}}\leq{42}\rightarrow{\left({1}\right)}\)
\(\displaystyle{15}{x}_{{1}}+{7}{x}_{{2}}\leq{38}\rightarrow{\left({2}\right)}\)
Now to equalize the above equation , we add \(\displaystyle{s}_{{1}}\ \text{ and }\ {s}_{{2}}\) to the above eqn (1) and (2) respectively,
\(\displaystyle{3}{x}_{{1}}+{9}{x}_{{2}}+{s}_{{1}}={42}\rightarrow{\left({1}{a}\right)}\)
\(\displaystyle{15}{x}_{{1}}+{7}{x}_{{2}}+{s}_{{2}}={38}\rightarrow{\left({2}{f}\right)}\)
Step 2
\(\displaystyle\therefore\) For initial simplex , basis feasible solution in \(\displaystyle{x}_{{1}}={0}\ \text{ and }\ {x}_{{2}}={0}\)
\(\displaystyle\Rightarrow{s}_{{1}}={42}\ \text{ and }\ {s}_{{2}}={38}\)
So, answer.
\(\displaystyle{3}{x}_{{1}}+{9}{x}_{{2}}+{s}_{{1}}={42}\)
\(\displaystyle{15}{x}_{{1}}+{7}{x}_{{2}}+{s}_{{2}}={38}\)