Chapter 5

A system for tracking ships indicates that a ship lies on a path described by \(2 y^{2}-x^{2}-1 .\) The process is repeated and the ship is found to lie on a path described by \(2 x^{2}-y^{2}-1\). If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.

Exercises \(55-57\) will help you prepare for the material covered in the next section. Solve: $$ \left\\{\begin{array}{r} A+B=3 \\ 2 A-2 B+C=17 \\ 4 A-2 C=14 \end{array}\right. $$

Explain how to find the partial fraction decomposition of a rational expression with a repeated, prime quadratic factor in the denominator.

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

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 3 x+y \leq 6 \\ x>-2 \\ y \leq 4 \end{array}\right.$$

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+5 y<10 \\ 3 x+4 y \leq 12 \end{array}\right.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Partial fraction decomposition involves finding a single rational expression for a given sum or difference of rational expressions.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+y<4 \\ 2 x-3 y \leq 6 \end{array}\right.$$

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000\). It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent he number of canoes produced and sold, )

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 3 x+y \leq 6 \\ 2 x-y \leq-1 \\ x>-2 \\ y<4 \end{array}\right.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$\frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2}$$

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures bicycles has a fixed cost of \(\$ 100,000\). It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{c} 2 x+y \leq 6 \\ x+y>2 \\ 1 \leq x \leq 2 \\ y<3 \end{array}\right.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125 .\) ( In solving this exercise, let \(x\) represent the number of sold-out performances.)

In Exercises 63–64, write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 4 more than the product of \(-2\) and the \(x\)-variable.

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost \(2 \notin\) per card and you are selling each card for \(50 \mathrm{e}\). (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

What is a system of nonlinear equations? Provide an example with your description.

In Exercises 63–64, write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 2 more than the product of \(-3\) and the \(x\)-variable.

Involve supply and demand. The following models describe wages for low-skilled labor. \(\begin{array}{lcc}\text { Demand Model } & \text { Supply Model } \\ p-- 0.325 x+5.8 & p-0.375 x+3\end{array}\) a. Solve the system and find the equilibrium number of workers, in millions, and the equilibrium hourly wage. b. Use your answer from part (a) to complete this statement: If workers are paid ___ per hour, there will be ___ million available workers and ___ millions workers will be hired. c. In 2007 , the federal minimum wage was set at \(\$ 5.15\) per hour. Substitute 5.15 for \(p\) in the demand model, \(p--0.325 x+5.8,\) and determine the millions of workers employers will hire at this price. d. At a minimum wage of \(\$ 5.15\) per hour, use the supply model, \(p-0.375 x+3,\) to determine the millions of available workers. Round to one decimal place. e. At a minimum wage of \(\$ 5.15\) per hour, use your answers from parts (c) and (d) to determine how many more people are looking for work than employers are willing to hire.

Find the partial fraction decomposition of $$\frac{4 x^{2}+5 x-9}{x^{3}-6 x-9}$$

Explain how to solve a nonlinear system using the substitution method. Use \(x^{2}+y^{2}-9\) and \(2 x-y-3\) to illustrate your explanation.

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is at most \(4 .\) The \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed 6.

Involve supply and demand. The following models describe demand and supply for three bedroom rental apartments. \(\begin{array}{lc}\text { Demand Model } & \text { Supply Model } \\ p--50 x+2000 & p-50 x\end{array}\) a. Solve the system and find the equilibrium quantity and the equilibrium price. b. Use your answer from part (a) to complete this statement: When rents are ___ per month, consumers will demand ___ apartments and suppliers will offer ___ appartments for rent.

Will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{l} 4 x+3 y=4 \\ y=2 x-7 \end{array}\right.$$

Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}-5\) and \(3 x^{2}-2 y^{2}-19\) to illustrate your explanation.

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is at most \(3 .\) The \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6.

Will help you prepare for the material covered in the next section. Solve by the addition method: $$ \left\\{\begin{array}{l} 2 x+4 y=-4 \\ 3 x+5 y=-3 \end{array}\right. $$

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is no more than 2. The \(y\)-variable is no less than the difference between the square of the \(x\)-variable and 4.

Will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the squares of the \(x\)-variable and the \(y\)-variable is no more than \(25.\) The sum of twice the \(y\)-variable and the \(x\)-variable is no less than 5.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the same steps to solve nonlinear systems as I did to solve linear systems, although I don't obtain linear equations when a variable is eliminated.

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$\left\\{\begin{array}{l} |x| \leq 2 \\ |y| \leq 3 \end{array}\right.$$

Involve supply and demand. Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In \(2000,\) the cost of Social Security was \(5.48 \%\) of the gross domestic product, increasing by \(0.04 \%\) of the GDP per year. In \(2000,\) the cost of Medicare was \(1.84 \%\) of the gross domestic product, increasing by \(0.17 \%\) of the GDP per year. a. Write a function that models the cost of Social Security as a percentage of the GDP \(x\) years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP \(x\) years after 2000 . c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a nonlinear system that modeled the orbits of Earth and Mars, and the graphs indicated the system had a solution with a real ordered pair.

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$\left\\{\begin{array}{l} |x| \leq 1 \\ |y| \leq 2 \end{array}\right.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Without using any algebra, it's obvious that the nonlinear system consisting of \(x^{2}+y^{2}-4\) and \(x^{2}+y^{2}-25\) does not have real-number solutions.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I think that the nonlinear system consisting of \(x^{2}+y^{2}-36\) and \(y-(x-2)^{2}-3\) is casier to solve graphically than by using the substitution method or the addition method.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are a circle and a line can have four real ordered-pair solutions

Without graphing, in Exercises 73–76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} 3 x+y<9 \\ 3 x+y>9 \end{array}\right.$$

Without graphing, in Exercises 73–76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} 6 x-y \leq 24 \\ 6 x-y>24 \end{array}\right.$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions

Without graphing, in Exercises 73–76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} (x+4)^{2}+(y-3)^{2} \leq 9 \\ (x+4)^{2}+(y-3)^{2} \geq 9 \end{array}\right.$$

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Basedon the graph. (GRAPH CAN'T COPY) A collection of Halloween candy contains a total of 12 Snickers bars and Reese's Peanut Butter Cups. Chew on this: The grams of fat in these candy bars exceed twice the daily maximum desirable fat intake of 70 grams by 26.5 grams. How many bars of each kind of candy are contained in the Halloween collection?

Without graphing, in Exercises 73–76, determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} (x-4)^{2}+(y+3)^{2} \leq 24 \\ (x-4)^{2}+(y+3)^{2} \geq 24 \end{array}\right.$$