Chapter 7

For the following exercises, perform the operation and then find the partial fraction decomposition. $$ \frac{7}{x+8}+\frac{5}{x-2}-\frac{x-1}{x^{2}-6 x-16} $$

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. A laptop company has discovered their cost and revenue functions for each day: \(C(x)=3 x^{2}-10 x+200\) and \(R(x)=-2 x^{2}+100 x+50\). If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $$\$ 82,000$$. The office manager makes $$\$ 4,000$$ more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $$\$ 78,000$$. What is the annual salary of each of the co-workers?

For the following exercises, solve for the desired quantity. A fast-food restaurant has a cost of production \(C(x)=11 x+120\) and \(a\) revenue function \(R(x)=5 x\). When does the company start to turn a profit?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. You sold two types of scarves at a farmers' market and would like to know which one was more popular. The total number of scarves sold was \(56,\) the yellow scarf cost \(\$ 10,\) and the purple scarf cost \(\$ 11\). If you had total revenue of \(\$ 583,\) how many yellow scarves and how many purple scarves were sold?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. Anna, Ashley, and Andrea weigh a combined \(370 \mathrm{lb}\). If Andrea weighs \(20 \mathrm{lb}\) more than Ashley, and Anna weighs 1.5 times as much as Ashley, how much does each girl weigh?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising \(83 \%\) of the flavors sold at an ice cream shop. If vanilla sells \(1 \%\) more than twice strawberry, and chocolate sells \(11 \%\) more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) $$ B^{5} $$

For the following exercises, perform the operation and then find the partial fraction decomposition. $$ \frac{1}{x-4}-\frac{3}{x+6}-\frac{2 x+7}{x^{2}+2 x-24} $$

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. A cell phone company has the following cost and revenue functions: \(C(x)=8 x^{2}-600 x+21,500\) and \(R(x)=-3 x^{2}+480 x .\) What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

At a carnival, $$\$ 2,914.25$$ in receipts were taken at the end of the day. The cost of a child's ticket was $$\$ 20.50,$$ an adult ticket was $$\$ 29.75,$$ and a senior citizen ticket was $$\$ 15.25$$. There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?

For the following exercises, solve for the desired quantity. A cell phone factory has a cost of production \(C(x)=150 x+10,000\) and \(a\) revenue function \(R(x)=200 x\). What is the break-even point?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of \(13 \mathrm{lb}\), 14 oz. How many of each type of tomato do you have?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up \(12 \%\) of total ice cream sales. This year, the same three ice creams made up \(16.9 \%\) of ice cream sales. The rocky road sales doubled, the banana sales increased by \(50 \%,\) and the pumpkin sales increased by \(20 \%\). If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

For the following exercises, perform the operation and then find the partial fraction decomposition. $$ \frac{2 x}{x^{2}-16}-\frac{1-2 x}{x^{2}+6 x+8}-\frac{x-5}{x^{2}-4 x} $$

A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $$\$ 28,112.50 .$$ The tickets were priced at $$\$ 20$$ for student tickets, $$\$ 22.50$$ for children, and $$\$ 29$$ for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

For the following exercises, solve for the desired quantity. A musician charges \(C(x)=64 x+20,000\) where \(x\) is the total number of attendees at the concert. The venue charges $$\$ 80$$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At a market, the three most popular vegetables make up \(53 \%\) of vegetable sales. Corn has \(4 \%\) higher sales than broccoli, which has \(5 \%\) more sales than onions. What percentage does each vegetable have in the market share?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost \(\$ 2\) per square foot, the wood \(\$ 10\) per square foot, and the plywood \(\$ 5\) per square foot. The farmer spent a total of \(\$ 51\) and the total amount of materials used was \(14 \mathrm{ft}^{2}\). He used \(3 \mathrm{ft}^{2}\) more chicken wire than plywood. How much of each material in did the farmer use?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh \(4 \mathrm{~g}\), and almonds weigh \(5 \mathrm{~g}\). If the bag weighs \(3.7 \mathrm{~kg},\) find out how many of each type of nut is in the bag.

In a bag, a child has 325 coins worth $$\$ 19.50$$. There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?

For the following exercises, solve for the desired quantity. A guitar factory has a cost of production \(C(x)=75 x+50,000 .\) If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At the same market, the three most popular fruits make up \(37 \%\) of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. \(30 \%\) of the almonds, \(20 \%\) of the cashews, and \(10 \%\) of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

Last year, at Haven's Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $$\$ 140,000$$. This year, due to inflation, the same cars would cost $$\$ 151,830$$. The cost of the BMW increased by \(8 \%,\) the Jeep by \(5 \%,\) and the Toyota by \(12 \%\). If the price of last year's Jeep was $$\$ 7,000$$ less than the price of last year's BMW, what was the price of each of the three cars last year?

For the following exercises, use a system of linear equations with two variables and two equations to solve. Find two numbers whose sum is 28 and difference is \(13 .\)

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Three bands performed at a concert venue. The first band charged \(\$ 15\) per ticket, the second band charged \(\$ 45\) per ticket, and the final band charged \(\$ 22\) per ticket. There were 510 tickets sold, for a total of \(\$ 12,700\). If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

For the following exercises, solve the system for \(x, y,\) and \(z\). A recent college graduate took advantage of his business education and invested in three investments immediately after graduating. He invested $$\$ 80,500$$ into three accounts, one that paid \(4 \%\) simple interest, one that paid \(3 \frac{1}{8} \% \quad\) simple interest, and one that paid \(2 \frac{1}{2} \%\) simple interest. He earned $$\$ 2,670$$ interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?

For the following exercises, use a system of linear equations with two variables and two equations to solve. A number is 9 more than another number. Twice the sum of the two numbers is 10 . Find the two numbers.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. A movie theatre sold tickets to three movies. The tickets to the first movie were \(\$ 5,\) the tickets to the second movie were \(\$ 11\), and the third movie was \(\$ 12\). 100 tickets were sold to the first movie. The total number of tickets sold was \(642,\) for a total revenue of \(\$ 6,774\). How many tickets for each movie were sold?

For the following exercises, use a system of linear equations with two variables and two equations to solve. The startup cost for a restaurant is $$\$ 120,000,$$ and each meal costs $$\$ 10$$ for the restaurant to make. If each meal is then sold for $$\$ 15,$$ after how many meals does the restaurant break even?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Men aged \(20-29,30-39,\) and 40-49 made up \(78 \%\) of the population at a prison last year. This year, the same age groups made up \(82.08 \%\) of the population. The \(20-29\) age group increased by \(20 \%,\) the 30-39 age group increased by \(2 \%,\) and the \(40-49\) age group decreased to \(\frac{3}{4}\) of their previous population. Originally, the \(30-39\) age group had \(2 \%\) more prisoners than the \(20-29\) age group. Determine the prison population percentage for each age group last year.

For the following exercises, solve the system for \(x, y,\) and \(z\). You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays \(4 \%\) compounded annually, the second account pays \(3 \%\) compounded annually, and the third account pays \(2 \%\) compounded annually. After one year, you earn $$\$ 3,650$$ in interest. If you invest five times the money in the account that pays \(4 \%\) compared to \(3 \%,\) how much did you invest in each account?

For the following exercises, use a system of linear equations with two variables and two equations to solve. A moving company charges a flat rate of $$\$ 150,$$ and an additional $$\$ 5$$ for each box. If a taxi service would charge $$\$ 20$$ for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At a women's prison down the road, the total number of inmates aged \(20-49\) totaled \(5,525 .\) This year, the \(20-29\) age group increased by \(10 \%,\) the 30-39 age group decreased by \(20 \%,\) and the \(40-49\) age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the \(30-39\) age group than the \(20-29\) age group. Determine the prison population for each age group last year.

For the following exercises, use a system of linear equations with two variables and two equations to solve. A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?

For the following exercises, use a system of linear equations with two variables and two equations to solve. 276 students enrolled in a freshman-level chemistry class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.

For the following exercises, use a system of linear equations with two variables and two equations to solve. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?

For the following exercises, solve the system for \(x, y,\) and \(z\). The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for \(64 \%\) of the United States oil production. The Gulf of Mexico and Texas combined for \(47 \%\) of oil production. Texas produced 3\% more than Alaska. What percent of United States oil production came from these regions?

For the following exercises, use a system of linear equations with two variables and two equations to solve. A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

For the following exercises, solve the system for \(x, y,\) and \(z\). At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised \(55 \%\) of the endangered species. Birds accounted for \(0.7 \%\) more than fish, and fish accounted for \(1.5 \%\) more than mammals. What percent of the endangered species came from mammals, birds, and fish?

For the following exercises, use a system of linear equations with two variables and two equations to solve. If a scientist mixed \(10 \%\) saline solution with \(60 \%\) saline solution to get 25 gallons of \(40 \%\) saline solution, how many gallons of \(10 \%\) and \(60 \%\) solutions were mixed?

For the following exercises, use a system of linear equations with two variables and two equations to solve. An investor earned triple the profits of what she earned last year. If she made $$\$ 500,000.48$$ total for both years, how much did she earn in profits each year?

For the following exercises, use a system of linear equations with two variables and two equations to solve. An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a \(110 \%\) increase on the money she invested. On the second investment, Riverside Community, she earned \(50 \%\) over what she invested. If she earned \(\$ 1\) million in profits, how much did she invest in each of the land deals?

For the following exercises, use a system of linear equations with two variables and two equations to solve. If an investor invests a total of $$\$ 25,000$$ into two bonds, one that pays \(3 \%\) simple interest, and the other that pays \(2 \frac{7}{8} \%\) interest, and the investor earns \(\$ 737.50\) annual interest, how much was invested in each account?

For the following exercises, use a system of linear equations with two variables and two equations to solve. If an investor invests $$\$ 23,000$$ into two bonds, one that pays \(4 \%\) in simple interest, and the other paying \(2 \%\) simple interest, and the investor earns $$\$ 710.00$$ annual interest, how much was invested in each account?

For the following exercises, use a system of linear equations with two variables and two equations to solve. CDs cost $$\$ 5.96$$ more than DVDS at All Bets Are Off Electronics. How much would \(6 \mathrm{CDs}\) and 2 DVDs cost if 5 CDs and 2 DVDS cost $$\$ 127.73 ?$$

For the following exercises, use a system of linear equations with two variables and two equations to solve. A store clerk sold 60 pairs of sneakers. The high-tops sold for $$\$ 98.99$$ and the low-tops sold for $$\$ 129.99 .$$ If the receipts for the two types of sales totaled $$\$ 6,404.40$$, how many of each type of sneaker were sold?

For the following exercises, use a system of linear equations with two variables and two equations to solve. A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $$\$ 12.50,$$ and the price for an adult ticket was $$\$ 16.00$$. The register confirms that $$\$ 5,075$$ was taken in. How many student tickets and adult tickets were sold?