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Introduce nonnegative slack variables $s_1,s_2,s_3$ and start the simplex algorithm at $x=y=z=0$, with slacks $9,8,2$. The objective's largest positive entering coefficient belongs to $y$. Its ratio test is $9/3=3$, $8/2=4$, $2/1=2$, so $s_3$ leaves and $y$ enters.

Solve that pivot row for $y$ and substitute into the other rows and the objective $v$:This is a feasible simplex dictionary at nonbasic variables $x=z=s_3=0$. All their objective coefficients are negative, so no feasible increase can improve the objective. ThusThe negative reduced coefficients also prove uniqueness of this optimum.

The dual linear program, with nonnegative variables $u_1,u_2,u_3$, isThe feasible choice $\boxed{(u_1,u_2,u_3)=(0,0,5)}$ has objective $10$. Weak duality proves that it and the primal point are optimal. Complementary slackness also forces $u_1=u_2=0$ because the first two primal slacks are positive, and then the positive $y$ forces $u_3=5$.

Under the null hypothesis the three outcome probabilities are the same in the two independent samples. The pooled row totals are $50,20,30$, and both column totals are $50$, out of a total of $100$. Thus the expected counts in each column are $25,10,15$. These exceed the usual small-count threshold for the asymptotic Pearson chi-squared test of homogeneity.

The Pearson chi-squared statistic for contingency tables isUnder the null, its asymptotic chi-squared distribution has $(3-1)(2-1)=2$ degrees of freedom. Equivalently, the two independent three-category samples have four probability parameters unrestricted and two under the common-probability null. The $1\%$ upper-tail critical value is $9.21$.

Since $9.12<9.21$, do not reject the equal-effect hypothesis at the 1% level. For two degrees of freedom, the approximate p-value is $e^{-9.12/2}\simeq 0.01046$, just above $0.01$. This is a decision at the specified significance level, not evidence that the two outcome distributions have been proved identical.