Chapter 7

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Three numbers add up to 106 . The first number is 3 less than the second number. The third number is 4 more than the first number.

For the following exercises, find the inverse of the given matrix. $$ \left[\begin{array}{llllll} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right] $$

For the following exercises, use Gaussian elimination to solve the system. $$ \begin{array}{l} \frac{x-3}{4}-\frac{y-1}{3}+2 z=-1 \\ \frac{x+5}{2}+\frac{y+5}{2}+\frac{z+5}{2}=7 \\ x+y+z=1 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(B A\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{5 x+2}{x\left(x^{2}+4\right)^{2}} $$

For the following exercises, find the solutions to the nonlinear equations with two variables. $$ \begin{array}{l} x^{2}-x y-2 y^{2}-6=0 \\ x^{2}+y^{2}=1 \end{array} $$

Three even numbers sum up to 108. The smaller is half the larger and the middle number is \(\frac{3}{4}\) the larger. What are the three numbers?

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). \(x+y=A\) \(x-y=B\)

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Three numbers add to 216 . The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(C A\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{x^{4}+x^{3}+8 x^{2}+6 x+36}{x\left(x^{2}+6\right)^{2}} $$

For the following exercises, find the solutions to the nonlinear equations with two variables. $$ \begin{array}{l} x^{2}+4 x y-2 y^{2}-6=0 \\ x=y+2 \end{array} $$

Three numbers sum up to 147 . The smallest number is half the middle number, which is half the largest number. What are the three numbers?

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). \(x+A y=1\) \(x+B y=1\)

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 invest \(\$ 10,000\) into two accounts, which receive \(8 \%\) interest and \(5 \%\) interest. At the end of a year, you had \(\$ 10,710\) in your combined accounts. How much was invested in each account?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. At a competing cupcake store, \(\$ 4,520\) worth of cupcakes are sold daily. The chocolate cupcakes cost \(\$ 2.25\) and the red velvet cupcakes cost \(\$ 1.75 .\) If the total number of cupcakes sold per day is \(2,200,\) how many of each flavor are sold each day?

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(B C\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{2 x-9}{\left(x^{2}-x\right)^{2}} $$

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer. $$ \begin{array}{l} x y<1 \\ y>\sqrt{x} \end{array} $$

At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). \(A x+y=0\) \(B x+y=1\)

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 invest \(\$ 80,000\) into two accounts, \(\$ 22,000\) in one account, and \(\$ 58,000\) in the other account. At the end of one year, assuming simple interest, you have earned \(\$ 2,470\) in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was \(348 \mathrm{lb}, 12\) oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. You invested \(\$ 10,000\) into two accounts: one that has simple \(3 \%\) interest, the other with \(2.5 \%\) interest. If your total interest payment after one year was \(\$ 283.50,\) how much was in each account after the year passed?

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(A B C\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{5 x^{3}-2 x+1}{\left(x^{2}+2 x\right)^{2}} $$

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer. $$ \begin{array}{l} x^{2}+y<3 \\ y>2 x \end{array} $$

An animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). \(A x+B y=C\) \(x+y=1\)

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 theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children's tickets are \(\$ 5.95,\) adult tickets are \(\$ 11.15,\) and the total amount of revenue was \(\$ 12,756,\) how many children's tickets and adult tickets 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. Students were asked to bring their favorite fruit to class. \(95 \%\) of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were \(5 \%\) less popular than bananas, what are the percentages of each individual fruit?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. You invested \(\$ 2,300\) into account \(1,\) and \(\$ 2,700\) into account 2 . If the total amount of interest after one year is \(\$ 254\) and account 2 has 1.5 times the interest rate of account \(1,\) what are the interest rates? Assume simple interest rates.

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) $$ B^{2} $$

For the following exercises, find the partial fraction expansion. $$ \frac{x^{2}+4}{(x+1)^{3}} $$

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. Two numbers add up to 300 . One number is twice the square of the other number. What are the numbers?

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $$\$ 82$$. She forgot to save the individual receipts but remembered that your groceries were $$\$ 0.05$$ cheaper than half of her groceries, and that your other roommate's groceries were $$\$ 2.10$$ more than your groceries. How much was each of your share of the groceries?

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). $$ \begin{array}{l} A x+B y=C \\ D x+E y=F \end{array} $$

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 concert venue sells single tickets for \(\$ 40\) each and couple's tickets for \(\$ 65 .\) If the total revenue was \(\$ 18,090\) and the 321 tickets were sold, how many single tickets and how many couple's tickets 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. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \(\$ 1\) and the chocolate chip cookies at \(\$ 0.75\). They raised \(\$ 700\) and sold 850 items. How many brownies and how many cookies were sold?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Bikes'R'Us manufactures bikes, which sell for \(\$ 250\). It costs the manufacturer \(\$ 180\) per bike, plus a startup fee of \(\$ 3,500\) After how many bikes sold will the manufacturer break even?

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) $$ B^{3} $$

For the following exercises, find the partial fraction expansion. $$ \frac{x^{3}-4 x^{2}+5 x+4}{(x-2)^{3}} $$

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. The squares of two numbers add to 360 . The second number is half the value of the first number squared. What are the numbers?

Your roommate, John, offered to buy household supplies for you and your other roommate. You live near the border of three states, each of which has a different sales tax. The total amount of money spent was $$\$ 100.75 .$$ Your supplies were bought with \(5 \%\) tax, John's with \(8 \%\) tax, and your third roommate's with \(9 \%\) sales tax. The total amount of money spent without taxes is $$\$ 93.50 .$$ If your supplies before tax were $$\$ 1$$ more than half of what your third roommate's supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.

For the following exercises, solve for the desired quantity. A stuffed animal business has a total cost of production \(C=12 x+30\) and a revenue function \(R=20 x\). Find 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. You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was \(\$ 29.50 .\) If each gallon of yellow costs \(\$ 2.59,\) and each gallon of blue costs \(\$ 3.19,\) how many gallons of each color go into your green mix?

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at \$13.99, the beanie at \(\$ 7.99\), and the cowboy hat at \(\$ 14.49 .\) If 100 hats were sold this past quarter, \(\$ 1,119\) was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for \(\$ 86\) each, with a delivery fee of \(\$ 9,200\), regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) $$ B^{4} $$