Chapter 9

A coffee company purchases mixed lots of coffee beans and then grades them into premium, regular, and unusable beans. The company needs at least 280 tons of premium-grade and 200 tons of regular-grade coffee beans. The company can purchase ungraded coffee from two suppliers in any amount desired. Samples from the two suppliers contain the following percentages of premium, regular, and unusable beans: $$ \begin{array}{|c|c|c|c|} \hline \text { Supplier } & \text { Premium } & \text { Regular } & \text { Unusable } \\ \hline \text { A } & 20 \% & 50 \% & 30 \% \\ \text { B } & 40 \% & 20 \% & 40 \% \\ \hline \end{array} $$ If supplier A charges $$\$ 900$$ per ton and B charges $$\$ 1200$$ per ton, how much should the company purchase from each supplier to fulfill its needs at minimum cost?

\(\left\\{\begin{array}{c}y-x<0 \\ 2 x+5 y<10\end{array}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ 3 x+4 y=-25 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{array}{l} 0.11 x-0.03 y=0.25 \\ 0.12 x+0.05 y=0.70 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{3 x^{3}+11 x^{2}+16 x+5}{x(x+1)^{3}} $$

If \(A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\), show that \(A I_{3}=A=I_{3} A\)

A farmer, in the business of growing fodder for livestock, has 90 acres available for planting alfalfa and corn. The cost of seed per acre is $$\$ 32$$ for alfalfa and $$\$ 48$$ for corn. The total cost of labor will amount to $$\$ 60$$ per acre for alfalfa and $$\$ 30$$ per acre for corn. The expected revenue (before costs are subtracted) is $$\$ 500$$ per acre from alfalfa and $$\$ 700$$ per acre from corn. If the farmer does not wish to spend more than $$\$ 3840$$ for seed and $$\$ 4200$$ for labor, how many acres of each crop should be planted to obtain the maximum profit?

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{rrr} 4 & -3 & 1 \\ -5 & 2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 0 & 1 \\ -4 & 7 \end{array}\right] $$

\(\left\\{\begin{aligned} 3 x+y & \leq 6 \\ y-2 x & \geq 1 \\ x & \geq-2 \\\ y & \leq 4 \end{aligned}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=9 \\ y-3 x=2 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{array}{r} 2 x-3 y=5 \\ -6 x+9 y=12 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{4 x^{3}+3 x^{2}+5 x-2}{x^{3}(x+2)} $$

Show that \(A I_{4}=A=I_{4} A\) for every square matrix \(A\) of order 4 .

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{rrrr} 2 & 1 & -1 & 0 \\ 3 & -2 & 0 & 5 \\ -2 & 1 & 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 5 & -3 & 1 \\ 1 & 2 & 0 \\ -1 & 0 & 4 \\ 0 & -2 & 3 \end{array}\right] $$

A small firm manufactures bookshelves and desks for microcomputers. For each product it is necessary to use a table saw and a power router. To manufacture each bookshelf, the saw must be used for \(\frac{1}{2}\) hour and the router for 1 hour. A desk requires the use of each machine for 2 hours. The profits are $$\$ 20$$ per bookshelf and $$\$ 50$$ per desk. If the saw can be used for 8 hours per day and the router for 12 hours per day, how many bookshelves and desks should be manufactured each day to maximize the profit?

\(\left\\{\begin{aligned} 3 x-4 y & \geq 12 \\ x-2 y & \leq 2 \\ x & \geq 9 \\\ y & \leq 5 \end{aligned}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=16 \\ y+2 x=-1 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{array}{r} 3 p-q=7 \\ -12 p+4 q=3 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{2}+x-6}{\left(x^{2}+1\right)(x-1)} $$

Exer. 17-20: Solve the system using the inverse method. Refer to Exercises 3-4 and 9-10. $$ \left\\{\begin{array}{r} 2 x-4 y=c \\ x+3 y=d \end{array}\right. $$ (a) \(\left[\begin{array}{l}c \\ d\end{array}\right]=\left[\begin{array}{l}3 \\\ 1\end{array}\right]\) (b) \(\left[\begin{array}{l}c \\ d\end{array}\right]=\left[\begin{array}{r}-2 \\\ 5\end{array}\right]\)

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$

Three substances, X, Y, and Z, each contain four ingredients, A, B, C, and D. The percentage of each ingredient and the cost in cents per ounce of each substance are given in the following table. $$ \begin{array}{|c|cccc|c|} \hline & \multicolumn{4}{|c|}{\text { Ingredients }} & \text { Cost per } \\ \cline { 2 - 5 } \text { Substance } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \text { ounce } \\ \hline \mathrm{X} & 20 \% & 10 \% & 25 \% & 45 \% & 25 \% \\ \mathrm{Y} & 20 \% & 40 \% & 15 \% & 25 \% & 35 \% \\ \mathrm{Z} & 10 \% & 20 \% & 25 \% & 45 \% & 50 \% \\ \hline \end{array} $$ If the cost is to be minimal, how many ounces of each substance should be combined to obtain a mixture of 20 ounces containing at least \(14 \% \mathrm{~A}, 16 \% \mathrm{~B}\), and \(20 \% \mathrm{C}\) ? What combination would make the cost greatest?

\(\left\\{\begin{aligned} 3 x-4 y & \geq 12 \\ x-2 y & \leq 2 \\ x & \geq 9 \\\ y & \leq 5 \end{aligned}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=16 \\ 2 y-x=4 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{aligned} 3 m-4 n &=2 \\ -6 m+8 n &=-4 \end{aligned}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{2}-x-21}{\left(x^{2}+4\right)(2 x-1)} $$

A man plans to operate a stand at a one-day fair at which he will sell bags of peanuts and bags of candy. He has $$\$ 2000$$ available to purchase his stock, which will cost $$\$ 2.00$$ per bag of peanuts and $$\$ 4.00$$ per bag of candy. He intends to sell the peanuts at $$\$ 3.00$$ and the candy at \(\$ 5.50\) per bag. His stand can accommodate up to 500 bags of peanuts and 400 bags of candy. From past experience he knows that he will sell no more than a total of 700 bags. Find the number of bags of each that he should have available in order to maximize his profit. What is the maximum profit?

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right] $$

\(\left\\{\begin{aligned} 2 x-3 y &=12 \\ 3 y+z &=-2 \\ 5 x-3 z &=3 \end{aligned}\right.\)

\(\left\\{\begin{array}{r}x+2 y \leq 8 \\ 0 \leq x \leq 4 \\ 0 \leq y \leq 3\end{array}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=1 \\ y+2 x=-3 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{array}{r} x-5 y=2 \\ 3 x-15 y=6 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{9 x^{2}-3 x+8}{x^{3}+2 x} $$

A small community wishes to purchase used vans and small buses for its public transportation system. The community can spend no more than $$\$ 100,000$$ for the vehicles and no more than $$\$ 500$$ per month for maintenance. The vans sell for $$\$ 10,000$$ each and average $$\$ 100$$ per month in maintenance costs. The corresponding cost estimates for each bus are $$\$ 20,000$$ and $$\$ 75$$ per month. If each van can carry 15 passengers and each bus can accommodate 25 riders, determine the number of vans and buses that should be purchased to maximize the passenger capacity of the system.

Find, if possible, \(A B\) and \(B A\). \(A=\left[\begin{array}{lll}-3 & 7 & 2\end{array}\right]\) $$ B=\left[\begin{array}{r} 1 \\ 4 \\ -5 \end{array}\right] $$

\(\left\\{\begin{array}{l}4 x-3 y=1 \\ 2 x+y=-7 \\ -x+y=-1\end{array}\right.\)

\(\left\\{\begin{array}{r}2 x+3 y \geq 6 \\ 0 \leq x \leq 5 \\ 0 \leq y \leq 4\end{array}\right.\)

$$ \left\\{\begin{aligned} (x-1)^{2}+(y+2)^{2} &=10 \\ x+y &=1 \end{aligned}\right. $$

Solve the system. $$ \left\\{\begin{array}{l} 2 y-5 x=0 \\ 3 y+4 x=0 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{2 x^{3}+2 x^{2}+4 x-3}{x^{4}+x^{2}} $$

Exer. 17-20: Solve the system using the inverse method. Refer to Exercises 3-4 and 9-10. $$ \left\\{\begin{aligned} x+2 y+3 z &=c \\ -2 x+y &=d \\ 3 x-y+z &=e \end{aligned}\right. $$ (a) \(\left[\begin{array}{l}c \\ d \\\ e\end{array}\right]=\left[\begin{array}{r}-1 \\ 4 \\ 2\end{array}\right]\) (b) \(\left[\begin{array}{l}c \\ d \\\ e\end{array}\right]=\left[\begin{array}{r}-3 \\ -2 \\ 1\end{array}\right]\)

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 4 & 8 \end{array}\right], \quad B=\left[\begin{array}{r} -3 \\ 2 \end{array}\right] $$

\(\left\\{\begin{array}{l}|x| \geq 2 \\ |y|<3\end{array}\right.\)

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{r} x y=2 \\ 3 x-y+5=0 \end{array}\right. $$

Solve the system. $$ \left\\{\begin{array}{r} 3 x+7 y=9 \\ y=5 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{4 x^{3}-x^{2}+4 x+2}{\left(x^{2}+1\right)^{2}} $$

\(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}c & d \\ a & b\end{array}\right| \quad \)

A fish farmer will purchase no more than 5000 young trout and bass from the hatchery and will feed them a special diet for the next year. The cost of food per fish will be \(\$ 0.50\) for trout and \(\$ 0.75\) for bass, and the total cost is not to exceed \(\$ 3000\). At the end of the year, a typical trout will weigh 3 pounds, and a bass will weigh 4 pounds. How many fish of each type should be stocked in the pond in order to maximize the total number of pounds of fish at the end of the year?

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{rrr} 2 & 0 & 1 \\ -1 & 2 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{array}\right] $$

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