91Ó°ÊÓ

Fill in the blanks. If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ________.

Interchanging two equations of a system of linear equations is a __________ ___________ that produces an equivalent system.

Fill in the blanks. You obtain the ________ ________ after multiplying each side of the partial fraction decomposition form by the least common denominator.

Determine whether each ordered triple is a solution of the system of equations. $$\left\\{\begin{array}{l} 3 x+4 y-z=17 \\ 5 x-y+2 z=-2 \\ 2 x-3 y+7 z=-21 \end{array}\right.$$ (a) (3,-1,2) (b) (1,3,-2) (c) (4,1,-3) (d) (1,-2,2)

Find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) Objective function: \(z=2 x+5 y\) Constraints: $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\ x+3 y \leq 15 \\\4 x+y \leq 16\end{array}$$

Find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) Objective function: \(z=10 x+7 y\) Constraints: $$\begin{aligned}0 \leq x \leq 60 \\\0 \leq y \leq 45 \\\5 x+6 y \leq 420\end{aligned}$$

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \(z=3 x+2 y\) Constraints: $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\\5 x+2 y \leq 20 \\\5 x+y \geq 10\end{array}$$

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \(z=5 x+\frac{1}{2} y\) Constraints: $$\begin{aligned}x & \geq 0 \\ y & \geq 0 \\\\\frac{1}{2} x+y & \leq 8 \\\x+\frac{1}{2} y & \geq 4\end{aligned}$$

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \(z=4 x+5 y\) Constraints: $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\\x+y \geq 8 \\\3 x+5 y \geq 30\end{array}$$

The linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: \(z=2.5 x+y\) Constraints: $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\ 3 x+5 y \leq 15 \\\5 x+2 y \leq 10\end{array}$$