Chapter 5

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}3 x+6 y \leq 6 \\\2 x+y \leq 8\end{array}\right.$$

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} y^{2}-x=4 \\ x^{2}+y^{2}=4 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x-4 y=11 \\ 2 x+3 y=-4 \end{array}\right. $$

Describe a situation in your life in which you would really like to maximize something, but you are limited by at least two constraints. Can linear programming be used in this situation? Explain your answer.

write the partial fraction decomposition of each rational expression. $$ \frac{x^{2}}{(x-1)^{2}(x+1)^{2}} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}3 x+6 y \leq 6 \\\2 x+y \leq 8\end{array}\right.$$

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x^{2}-2 y=8 \\ x^{2}+y^{2}=16 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 2 x+3 y=-16 \\ 5 x-10 y=30 \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to solve a linear programming problem, I use the graph representing the constraints and the graph of the objective function.

write the partial fraction decomposition of each rational expression. $$ \frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 2 x-5 y \leq 10 \\ 3 x-2 y>6 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} 3 x^{2}+4 y^{2}=16 \\ 2 x^{2}-3 y^{2}=5 \end{array}\right. $$

In Exercises 29-30, solve each system for \((x, y, z)\) in terms of the nonzero constants \(a, b,\) and \(c .\) $$\left\\{\begin{aligned} a x-b y-2 c z &=21 \\ a x+b y+c z &=0 \\ 2 a x-b y+c z &=14 \end{aligned}\right.$$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x=4 y+1 \\ 3 y=1-4 x \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the coordinates of each vertex from my graph representing the constraints to find the values that maximize or minimize an objective function.

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} 2 x-y \leq 4 \\ 3 x+2 y>-6 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x+y^{2}=4 \\ x^{2}+y^{2}=16 \end{array}\right. $$

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

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 5 x=6 y+40 \\ 2 y=8-3 x \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I need to be able to graph systems of linear inequalities in order to solve linear programming problems.

write the partial fraction decomposition of each rational expression. $$ \frac{5 x^{2}+6 x+3}{(x+1)\left(x^{2}+2 x+2\right)} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} y>2 x-3 \\ y<-x+6 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} 2 x^{2}+y^{2}=18 \\ x y=4 \end{array}\right. $$

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{cc} \hline x, \text { seconds after the ball is } & y, \text { ball's height, in feet, above } \\ \text { thrown } & \text { the ground } \\ \hline 1 & 224 \\ 3 & 176 \\ 4 & 104 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} x=9-2 y \\ x+2 y=13 \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. An important application of linear programming for businesses involves maximizing profit.

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} y<-2 x+4 \\ y

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=20 \\ x y=4 \end{array}\right. $$

A mathematical model can be used to describe the relationship between the number of feet a car travels once the brakes are applied, \(y,\) and the number of seconds the car is in motion after the brakes are applied, \(x .\) A research firm collects the following data: $$\begin{array}{cc} \hline \begin{array}{c} x \text { , seconds in motion } \\ \text { after brakes are applied } \end{array} & \begin{array}{c} y, \text { feet car travels } \\ \text { once the brakes are applied } \end{array} \\ \hline 1 & 46 \\ 2 & 84 \\ 3 & 114 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=6 .\) Describe what this means.

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} 6 x+2 y=7 \\ y=2-3 x \end{array}\right. $$

Use a system of linear equations in three variables to solve In this exercise, we refer to annual spending per person in 2010 dollar. The combined spending on housing, vehicles/gas, and health care was 3,840 dollar. The difference between spending on housing and spending on vehicles/gas was 3864 dollar. The difference between spending on housing and spending on health care was 695 dollar. Find the average per-person spending on housing, vehicles/gas, and health care in 2010 . (BAR GRAPH CAN'T COPY)

Suppose that you inherit 10,000 dollar. The will states how you must invest the money. Some (or all) of the money must be invested in stocks and bonds. The requirements are that at least 3000 dollar be invested in bonds, with expected returns of 0.08 dollar per dollar, and at least 2000 dollar be invested in stocks, with expected returns of 0.12 dollar per dollar. Because the stocks are medium risk, the final stipulation requires that the investment in bonds should never be less than the investment in stocks. How should the money be invested so as to maximize your expected returns?

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x+2 y \leq 4 \\ y \geq x-3 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=20 \\ x+2 y=6 \end{array}\right. $$

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} y=3 x-5 \\ 21 x-35=7 y \end{array}\right. $$

In this exercise, we refer to annual spending per person in 1980 . The combined spending on housing, vehicles/gas, and health care was 7073 dollar. The difference between spending on housing and spending on vehicles/gas was 1247 dollar. The difference between spending on housing and spending on health care was 1466 dollar. Find the average per-person spending on housing, vehicles/gas, and health care in 1980.

Consider the objective function \(z=A x+B y \quad(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9, x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the bbjective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A=\frac{2}{3} B\)

write the partial fraction decomposition of each rational expression. $$ \frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x+y \leq 4 \\ y \geq 2 x-4 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} 3 x^{2}-2 y^{2}=1 \\ 4 x-y=3 \end{array}\right. $$

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} 9 x-3 y=12 \\ y=3 x-4 \end{array}\right. $$

On a recent trip to the convenience store, you picked up 2 gallons of milk, 5 bottles of water, and 6 snack-size bags of chips. Your total bill (before tax) was 19.00 dollar. If a bottle of water costs twice as much as a bag of chips, and a gallon of milk costs 2.00 dollar more than a bottle of water, how much does each item cost?

Group members should choose a particular field of interest. Research how linear programming is used to solve problems in that field. If possible, investigate the solution of a specific practical problem. Present a report on your findings, including the contributions of George Dantzig, Narendra Karmarkar, and L. G. Khachion to linear programming.

write the partial fraction decomposition of each rational expression. $$ \frac{6 x^{2}-x+1}{x^{3}+x^{2}+x+1} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x \leq 2 \\ y \geq-1 \end{array}\right. $$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x^{3}+y=0 \\ x^{2}-y=0 \end{array}\right. $$

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} 3 x-2 y=-5 \\ 4 x+y=8 \end{array}\right. $$