Chapter 5

On a recent trip to the convenience store, you picked up 1 gallon of milk, 7 bottles of water, and 4 snack-size bags of chips. Your total bill (before tax) was dollar 17.00 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?

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 nethods used in conjunction with linear programming? What are these methods? What sort of academic background, garticularly in mathematics, does this executive have? Present group report addressing these questions, emphasizing the ole of linear programming for the business.

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x \leq 3 \\\y \leq-1\end{array}\right.$$

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

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

At a college production of \(A\) Streetcar Named Desire, 400 tickets were sold. The ticket prices were 8 dollar ,10 dollar, and 12 dollar, and the total income from ticket sales was 3700 dollar. How many tickets of each type were sold if the combined number of 8 dollar and 10 dollar tickets sold was 7 times the number of 12 dollar tickets sold?

Solve the system: $$\left\\{\begin{aligned} x+y+2 z &=19 \\ y+2 z &=13 \\ z &=5 \end{aligned}\right.$$ What makes it fairly easy to find the solution?

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2 \leq x<5$$

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x^{2}+(y-2)^{2}=4 \\ x^{2}-2 y=0 \end{array}\right. $$

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. $$ \left\\{\begin{array}{r} x+3 y=2 \\ 3 x+9 y=6 \end{array}\right. $$

A certain brand of razor blades comes in packages of \(6,12\) and 24 blades, costing 2 dollar , 3 dollar, and 4 dollar per package, respectively. A store sold 12 packages containing a total of 162 razor blades and took in 35 dollar. How many packages of each type were sold?

Solve the system: $$\begin{aligned} w-x+2 y-2 z &=-1 \\ x-\frac{1}{3} y+z &=\frac{8}{3} \\ y-z &=1 \\ z &=3 \end{aligned}$$ Express the solution set in the form \(\\{(w, x, y, z)\\} .\) What makes it fairly easy to find the solution?

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x^{2}-y^{2}-4 x+6 y-4=0 \\ x^{2}+y^{2}-4 x-6 y+12=0 \end{array}\right. $$

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

A person invested 6700 dollar for one year, part at 8%, part at 10%, and the remainder at 12% .$ The total annual income from these investments was 716 dollar. The amount of money invested at 12% was 300 dollar more than the amount invested at 8% and 10% combined. Find the amount invested at each rate.

Consider the following array of numbers:$$\left[\begin{array}{ccc}1 & 2 & -1 \\\ 4 & -3 &-15\end{array}\right]$$ 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. $$ \frac{x^{3}-4 x^{2}+9 x-5}{\left(x^{2}-2 x+3\right)^{2}} $$

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

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

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. $$ \left\\{\begin{array}{l} \frac{x}{4}-\frac{y}{4}=-1 \\ x+4 y=-9 \end{array}\right. $$

A person invested 17,000 dollar for one year, part at 10%, part at 12%, and the remainder at 15%.The total annual income from these investments was 2110 dollar . The amount of money invested at 12% was 1000 dollar 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. $$ \frac{3 x^{3}-6 x^{2}+7 x-2}{\left(x^{2}-2 x+2\right)^{2}} $$

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

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}=5 \\ 2 x-y=3 \end{array}\right. $$

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. $$ \left\\{\begin{array}{l} \frac{x}{6}-\frac{y}{2}=\frac{1}{3} \\ x+2 y=-3 \end{array}\right. $$

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x+y>4 \\\x+y<-1\end{array}\right.$$

In Exercises \(29-42,\) solve each system by the method of your choice.$$ \left\\{\begin{array}{l} x^{2}+y^{2}+3 y=22 \\ 2 x+y=-1 \end{array}\right. $$

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. $$ \left\\{\begin{array}{l} 2 x=3 y+4 \\ 4 x=3-5 y \end{array}\right. $$

What is a system of linear equations in three variables?

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

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

In Exercises \(29-42,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} x-3 y=-5 \\ x^{2}+y^{2}-25=0 \end{array}\right. $$

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. $$ \left\\{\begin{array}{l} 4 x=3 y+8 \\ 2 x=-14+5 y \end{array}\right. $$

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. $$ \frac{x^{5}+2}{x^{2}-1} $$

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x+y>4 \\\x+y>-1\end{array}\right.$$

let \(x\) represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is \(24 .\) Find the numbers.

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is \(7 .\) If one number is subtracted from the other, their difference is \(-1 .\) Find the numbers.

Describe in general terms how to solve a system in three variables.

perform each long division and write the partial fraction decomposition of the remainder term. $$ \frac{x^{5}}{x^{2}-4 x+4} $$

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

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is \(2 .\) If one number is subtracted from the other, their difference is \(8 .\) Find the numbers.

perform each long division and write the partial fraction decomposition of the remainder term. $$ \frac{x^{4}-x^{2}+2}{x^{3}-x^{2}} $$

let \(x\) represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is \(3 .\) Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

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