For each system, determine the number of slack variables and the size of the matrix for the initial simplex tableau.

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Maximize $$P = 9 x_1 + 5 x_2 + 9 x_3$$ subject to the constraints

$$\begin{array}{rrrrrcr} x_1 & - & x_2& + & x_3 & \le & 13 \cr 7 x_1 & & & + & x_3 & \le & 14 \cr x_1 & + & 12 x_2& - & 12 x_3 & \le & 14 \cr && &&x_{2} & \le & 14 \end{array}$$
$$x_1\ge 0 \qquad x_2 \ge 0 \qquad x_{3} \ge 0$$
How many slack variables for this system?

The matrix for the initial simplex tableau has rows and columns.

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Maximize $$P = 4 x_1 + 10 x_2$$ subject to the constraints

$$\begin{array}{rrrcr} 4 x_1 & + & x_2 & \le & 13 \cr 15 x_1 &- &x_2 &\le &77 \cr x_1\ge 0 &&x_2 & \ge & 0 \end{array}$$
How many slack variables for this system?

The matrix for the initial simplex tableau has rows and columns.

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Maximize $$P = 9 x_1 + x_2 + 5x_3$$ subject to the constraints

$$\begin{array}{rrrrrcr} 12 x_1 & - & x_2& + & 7 x_3 & \le & 10 \end{array}$$
$$x_1\ge 0 \qquad x_2 \ge 0 \qquad x_{3} \ge 0$$
How many slack variables for this system?

The matrix for the initial simplex tableau has rows and columns.