91Ó°ÊÓ

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x=4 y-2} \\ {x=6 y+8}\end{array}\right.\)

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$\begin{array}{lll} {} & {\text { Model } A} & {\text { Model } B} \\ {\text { Assembling }} & {5} & {4} \\ {\text { Painting }} & {2} & {3} \end{array}$$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are 25 for model A and 15 for model B . How many of each type should be produced to maximize profit?

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A large institution is preparing lunch menus containing foods A and \(\mathrm{B}\). The specifications for the two foods are given in the following table: $$\begin{array}{cccc} {} & {} & {\text { Units of }} & {\text { Units of }} \\ {} & {\text { Units of Fat }} & {\text { Carbohydrates }} & {\text { Protein }} \\ {\text { Food }} & {\text { per Ounce }} & {\text { per Ounce }} & {\text { per Ounce }} \\ {\mathrm{A}} & {1} & {2} & {1} \\ {\mathrm{B}} & {1} & {1} & {1} \end{array}$$ Each lunch must provide at least 6 units of fat per serving. no more than 7 units of protein, and at least 10 units of carbohydrates. The institution can purchase food A for 0.12 per ounce and food B for 0.08 per ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the least cost?

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{ll} {} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\ {\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\ {\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }} \end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to 35,000. \(\cdot\) Total payload had to be at least 672,000 pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

Solve each system for \((x, y, z)\) in terms of the nonzero constants \(a, b,\) and \(c .\) $$ \left\\{\begin{aligned} a x-b y+2 c z &=-4 \\ a x+3 b y-c z &=1 \\ 2 a x+b y+3 c z &=2 \end{aligned}\right. $$

Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {x^{2}+4 y^{2}=20} \\ {x+2 y=6} \end{array}\right. $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{c} {4 x-5 y \geq-20} \\ {x \geq-3} \end{array}\right. $$

perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{5}+2}{x^{2}-1}$$

The sum of two numbers is 7 . If one number is subtracted from the other, their difference is \(-1 .\) Find the numbers.