Chapter 5

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. \(\left\\{\begin{array}{l}x-9-2 y \\ x+2 y-13\end{array}\right.\)

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

In Exercises 27–62, 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

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

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. \(\left\\{\begin{array}{l}6 x+2 y-7 \\ y-2-3 x\end{array}\right.\)

Suppose that you inherit \(\$ 10,000 .\) 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 \(\$ 3000\) be invested in bonds, with expected returns of \(\$ 0.08\) per dollar, and at least \(\$ 2000\) be invested in stocks, with expected returns of \(\$ 0.12\) 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. $$\frac{x+4}{x^{2}\left(x^{2}+4\right)}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+2 y \leq 4 \\ y \geq x-3 \end{array}\right.$$

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

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. \(\left\\{\begin{array}{l}y-3 x-5 \\ 21 x-35-7 y\end{array}\right.\)

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 objective 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. $$\frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)}$$

Use a system of linear equations in three variables to solve Exercises \(33-4 I\) The bar graph shows the average annual spending per person on selected items in 1980 and \(2010 .\) All dollar amounts are adjusted for inflation. Use this display to solve Exercises \(33-34\) In this exercise, we refer to annual spending per person in \(1980 .\) The combined spending on housing, vehicles/gas, and health care was \(\$ 7073 .\) The difference between spending on housing and spending on vehicles/gas was \(\$ 1247 .\) The difference between spending on housing and spending on health care was \(\$ 1466 .\) Find the average per-person spending on housing. vehicles/gas, and health care in \(1980 .\)

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+y \leq 4 \\ y \geq 2 x-4 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 3 x^{2}-2 y^{2}-1 \\ 4 x-y-3 \end{array}\right.$$

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. \(\left\\{\begin{array}{l}9 x-3 y-12 \\ y-3 x-4\end{array}\right.\)

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. $$\frac{6 x^{2}-x+1}{x^{3}+x^{2}+x+1}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \leq 2 \\ y \geq-1 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{3}+y-0 \\ x^{2}-y-0 \end{array}\right.$$

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics, does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \leq 3 \\ y \leq-1 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{c} x^{3}+y-0 \\ 2 x^{2}-y-0 \end{array}\right.$$

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. \(\left\\{\begin{array}{l}2 x+5 y--4 \\ 3 x-y-11\end{array}\right.\)

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

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2 \leq x<5$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{2}+(y-2)^{2}-4 \\ x^{2}-2 y-0 \end{array}\right.$$

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. \(\left\\{\begin{array}{c}x+3 y-2 \\ 3 x+9 y-6\end{array}\right.\)

Exercises 37-39 will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$\left\\{\begin{aligned}w-x+2 y-2 z &=-1 \\\x-1 y+z &=1 \\\y-z &=1 \\\z-&-3\end{aligned}\right.$$ Express the solution set in the form \(\\{(\boldsymbol{x}, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z})\\} .\) What makes it fairly easy to find the solution?

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}+2 x+3}{\left(x^{2}+4\right)^{2}}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{2}-y^{2}-4 x+6 y-4=0 \\ x^{2}+y^{2}-4 x-6 y+12=0 \end{array}\right.$$

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. \(\left\\{\begin{array}{l}4 x-2 y-2 \\ 2 x-y-1\end{array}\right.\)

Exercises 37-39 will help you prepare for the material covered in the first section of the next chapter. Consider the following array of numbers: $$\left[\begin{array}{rrr}1 & 2 & -1 \\ 4 & -3 & -15\end{array}\right]$$ Rewrite the array as follows: Multiply each number in the top row by \(-4\) and add this product to the corresponding number in the bottom row. Do not change the numbers in the top row.

Write the partial fraction decomposition of each rational expression. $$\frac{x^{3}-4 x^{2}+9 x-5}{\left(x^{2}-2 x+3\right)^{2}}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x-y \leq 1 \\ x \geq 2 \end{array}\right.$$

Use a system of linear equations in three variables to solve Exercises \(33-4 I\) The bar graph shows the average annual spending per person on selected items in 1980 and \(2010 .\) All dollar amounts are adjusted for inflation. Use this display to solve Exercises \(33-34\) A person invested \(\$ 17,000\) for \(\overline{\text { oge year. part at } 10 \%, \text { part at }}\) \(12 \%,\) and the remainder at \(15 \% .\) The total annual income from these investments was \(\$ 240 .\) The amount of money invested at \(12 \%\) was \(\$ 1000\) less than the amount invested at \(10 \%\) and \(15 \%\) combined. Find the amount invested at each rate.

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{3}-6 x^{2}+7 x-2}{\left(x^{2}-2 x+2\right)^{2}}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{c} 4 x-5 y \geq-20 \\ x \geq-3 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}-5 \\ 2 x-y-3 \end{array}\right.$$

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}+3 x+14}{x^{3}-8}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+y>4 \\ x+y<-1 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{2}+y^{2}+3 y-22 \\ 2 x+y=-1 \end{array}\right.$$

Writing in Mathematics What is a system of linear equations in three variables?

Write the partial fraction decomposition of each rational expression. $$\frac{3 x-5}{x^{3}-1}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+y>3 \\ x+y<-2 \end{array}\right.$$

How do you determine whether a given ordered triple is a solution of a system in three variables?

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

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+y>4 \\ x+y>-1 \end{array}\right.$$