Chapter 11

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} x+y+z+w =0 \\ 2 x +w=0 \\ y-z =0 \\ x+2 z =1 \end{aligned}\right. $$

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{l}{y \geq x-3} \\ {y \geq-2 x+6} \\ {y \leq 8}\end{array}\right. $$

Nutrition A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce. (a) Find the inverse of the matrix $$\left[\begin{array}{lll}{3} & {1} & {3} \\ {4} & {2} & {4} \\ {3} & {2} & {4}\end{array}\right]$$ and use it to solve the remaining parts of this problem. (b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 10 \(\mathrm{mg}\) of folic acid, 14 \(\mathrm{mg}\) of choline, and 13 \(\mathrm{mg}\) of inositol? (c) How much of each food is needed to supply 9 \(\mathrm{mg}\) of folic acid, 12 \(\mathrm{mg}\) of choline, and 10 \(\mathrm{mg}\) of inositol? (d) Will any combination of these foods supply 2 \(\mathrm{mg}\) of folic acid, 4 \(\mathrm{mg}\) of choline, and 11 \(\mathrm{mg}\) of inositol?

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{4} y &=2 \\\\-8 x+6 y &=10 \end{aligned}\right. $$

Recognizing Partial Fraction Decompositions For each expression, determine whether it is already a partial fraction decomposition or whether it can be decomposed further. $$ \begin{array}{ll}{\text { (a) } \frac{x}{x^{2}+1}+\frac{1}{x+1}} & {\text { (b) } \frac{x}{(x+1)^{2}}} \\ {\text { (c) } \frac{1}{x+1}+\frac{2}{(x+1)^{2}}} & {\text { (d) } \frac{x+2}{\left(x^{2}+1\right)^{2}}}\end{array} $$

Can a Linear System Have Exactly Two Solutions? (a) Suppose that \(\left(x_{0}, y_{0}, z_{0}\right)\) and \(\left(x_{1}, y_{1}, z_{1}\right)\) are solutions of the system $$ \begin{array}{l}{\qquad\left\\{\begin{array}{l}{a_{1} x+b_{1} y+c_{1} z=d_{1}} \\\ {a_{2} x+b_{2} y+c_{2} z=d_{2}} \\ {a_{3} x+b_{3} y+c_{3} z=d_{3}}\end{array}\right.} \\ {\text { Show that }\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right) \text { is also a solution. }}\end{array} $$ (b) Use the result of part (a) to prove that if the system has two different solutions, then it has infinitely many solutions.

Solve the system of linear equations. $$ \left\\{\begin{array}{rr}{x+y-z-w=} & {6} \\ {2 x+\quad z-3 w=} & {8} \\\ {x-y \quad+4 w=} & {-10} \\ {3 x+5 y-z-w=} & {20}\end{array}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{x+y=1} \\ {y+z=2} \\ {z+w=3} \\\ {w-x=4}\end{array}\right. $$

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{aligned} x+y & \geq 12 \\ 2 x+y & \leq 24 \\ x-y & \geq-6 \end{aligned}\right. $$

Intersection of a Parabola and a Line On a sheet of graph paper or using a graphing calculator, draw the parabola \(y=x^{2} .\) Then draw the graphs of the linear equation \(y=x+k\) on the same coordinate plane for various values of \(k\) . Try to choose values of \(k\) so that the line and the parabola intersect at two points for some of your \(K^{\prime} s\) and not for others. For what value of \(k\) is there exactly one intersection point? Use the results of your experiment to make a conjecture about the values of \(k\) for which the following system has two solutions, one solution, and no solution. Prove your conjecture. $$ \left\\{\begin{array}{l}{y=x^{2}} \\ {y=x+k}\end{array}\right. $$

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{aligned}-\frac{1}{10} X+\frac{1}{2} y &=4 \\ 2 x-10 y &=-80 \end{aligned}\right. $$

Assembling and Disassembling Partial Fractions The following expression is a partial fraction decomposition. $$ \frac{2}{x-1}+\frac{1}{(x-1)^{2}}+\frac{1}{x+1} $$ Use a common denominator to combine the terms into one fraction. Then use the techniques of this section to find its partial fraction decomposition. Did you get back the original expression?

Produce Sales A farmer's three children, Amy, Beth, and Chad, run three roadside produce stands during the summer months. One weekend they all sell watermelons, yellow squash, and tomatoes. The matrices \(A\) and \(B\) tabulate the number of pounds of each product sold by each sibling on Saturday and Sunday. The matrix \(C\) gives the price per pound (in dollars) for each type of produce that they sell. Perform each of the following matrix operations, and interpret the entries in each result. \(\begin{array}{llll}{\text { (a) } A C} & {\text { (b) } B C} & {\text { (c) } A+B} & {\text { (d) }(A+B) C}\end{array}\)

Solve the system of linear equations. $$ \left\\{\begin{array}{rr}{x+y+2 z-w=} & {-2} \\ {3 y+z+2 w=} & {2} \\ {x+y} & {+3 w=2} \\ {-3 x} & {+z+2 w=} & {5}\end{array}\right. $$

Evaluate the determinants. $$ \left|\begin{array}{ccccc}{a} & {0} & {0} & {0} & {0} \\ {0} & {b} & {0} & {0} & {0} \\ {0} & {0} & {c} & {0} & {0} \\ {0} & {0} & {0} & {d} & {0} \\\ {0} & {0} & {0} & {0} & {e}\end{array}\right| $$

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{c}{y \leq 6 x-x^{2}} \\ {x+y \geq 4}\end{array}\right. $$

Sales Commissions An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set that she sells, she earns a commission based on the set's binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, earning \(\$ 625\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and leather sets, respectively. Translate the given information into a system of equations in \(X, y,\) and \(z .\) (b) Express the system of equations you found in part (a) as a matrix equation of the form \(A X=B\) . (c) Find the inverse of the coefficient matrix \(A\) and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on a set of encyclopedias in each grade of binding?

Some Trickier Systems Follow the hints and solve the systems. $$ \text { (a) }\left\\{\begin{array}{cc}{\log x+\log y} & {=\frac{3}{2}} \\ {2 \log x-\log y} & {=0}\end{array} \quad[\text { Hint: Add the equations. }]\right. $$ $$ \text { (b) }\left\\{\begin{array}{l}{2^{x}+2^{y}=10} \\\ {4^{x}+4^{y}=68}\end{array} \quad\left[\text { Hint: Note that } 4^{x}=2^{2 x}=\left(2^{x}\right)^{2}\right]\right. $$ $$ \text { (c) }\left\\{\begin{array}{cc}{x-y=3} & {\text { IHint: Factor the left-hand side }} \\ {x^{3}-y^{3}=387} & {\text { of the second equation. } ]}\end{array}\right. $$ $$ \text { (d) }\left\\{\begin{array}{ll}{x^{2}+x y=1} & {\text { [Hint: Add the equations, and }} \\ {x y+y^{2}=3} & {\text { factor the result. } ]}\end{array}\right. $$

\(49-52\) Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{array}{l}{0.21 x+3.17 y=9.51} \\ {2.35 x-1.17 y=5.89}\end{array}\right. $$

Evaluate the determinants. $$ \left|\begin{array}{ccccc}{a} & {a} & {a} & {a} & {a} \\ {0} & {a} & {a} & {a} & {a} \\ {0} & {0} & {a} & {a} & {a} \\ {0} & {0} & {0} & {a} & {a} \\\ {0} & {0} & {0} & {0} & {a}\end{array}\right| $$

Solve the system of linear equations. $$ \left\\{\begin{aligned} x-3 y+2 z+w &=-2 \\ x-2 y &-2 w=&-10 \\ z+5 w =15 \\\ 3 x &+2 z+w=-3 \end{aligned}\right. $$

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{l}{y \geq x^{3}} \\ {2 x+y \geq 0} \\ {y \leq 2 x+6}\end{array}\right. $$

No Zero-Product Property for Matrices We have used the Zero-Product Property to solve algebraic equations. Matrices do not have this property. Let \(O\) represent the \(2 \times 2\) zero matrix $$ O=\left[\begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right] $$ Find \(2 \times 2\) matrices \(A \neq O\) and \(B \neq O\) such that \(A B=O\) . Can you find a matrix \(A \neq O\) such that \(A^{2}=O ?\)

\(49-52\) Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{aligned} 18.72 x-14.91 y &=12.33 \\ 6.21 x-12.92 y &=17.82 \end{aligned}\right. $$

When Are Both Products Defined? What must be true about the dimensions of the matrices \(A\) and \(B\) if both products \(A B\) and \(B A\) are defined?

Solve for \(x\) $$ \left|\begin{array}{ccc}{x} & {12} & {13} \\ {0} & {x-1} & {23} \\ {0} & {0} & {x-2}\end{array}\right|=0 $$

Publishing Books A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

\(49-52\) Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{array}{l}{2371 x-6552 y=13,591} \\ {9815 x+992 y=618,555}\end{array}\right. $$

Powers of a Matrix Let $$ A=\left[\begin{array}{ll}{1} & {1} \\ {0} & {1}\end{array}\right] $$ Calculate \(A^{2}, A^{3}, A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n} .\)

Solve for \(x\) $$ \left|\begin{array}{lll}{x} & {1} & {1} \\ {1} & {1} & {x} \\ {x} & {1} & {x}\end{array}\right|=0 $$

Furniture Manufacturing \(\quad\) A man and his daughter manufacture unfinished tables and chairs. Each table requires 3 hours of sawing and 1 hour of assembly. Each chair requires 2 hours of sawing and 2 hours of assembly. Between the two of them, they can put in up to 12 hours of sawing and 8 hours of assembly work each day. Find a system of inequalities that describes all possible combinations of tables and chairs that they can make daily. Graph the solution set.

\(49-52\) Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{aligned}-435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right. $$

Powers of a Matrix Let \(A=\left[\begin{array}{cc}{1} & {1} \\ {1} & {1}\end{array}\right]\) Calculate \(A^{2}\) \(A^{3}, A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n} .\)

Solve for \(x\) $$ \left|\begin{array}{lll}{1} & {0} & {x} \\ {x^{2}} & {1} & {0} \\ {x} & {0} & {1}\end{array}\right|=0 $$

Solve the system of linear equations. $$ \left\\{\begin{aligned} x+z+w &=4 \\ y-z &=-4 \\ x-2 y+3 z+w &=12 \\ 2 x &-2 z+5 w=-1 \end{aligned}\right. $$

Coffee Blends \(A\) coffee merchant sells two different coffee blends. The Standard blend uses 4 oz of arabica and 12 oz of robusta beans per package; the Deluxe blend uses 10 oz of arabica and 6 oz of robusta beans per package. The merchant has 80 lb of arabica and 90 lo of robusta beans available. Find a system of inequalities that describes the possible number of Standard and Deluxe packages the merchant can make. Graph the solution set.

\(53-56\) Find \(x\) and \(y\) in terms of \(a\) and \(b\). $$ \left\\{\begin{array}{l}{x+y=0} \\ {x+a y=1}\end{array}(a \neq 1)\right. $$

Square Roots of Matrices A square root of a matrix \(B\) is a matrix \(A\) with the property that \(A^{2}=B .\) (This is the same definition as for a square root of a number.) Find as many square roots as you can of each matrix: $$ \left[\begin{array}{ll}{4} & {0} \\ {0} & {9}\end{array}\right] \quad\left[\begin{array}{ll}{1} & {5} \\ {0} & {9}\end{array}\right] $$ [Hint: If \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right],\) write the equations that \(a, b, c,\) and \(d\) would have to satisfy if \(A\) is the square root of the given matrix.

Solve for \(x\) $$ \left|\begin{array}{ccc}{a} & {b} & {x-a} \\ {x} & {x+b} & {x} \\ {0} & {1} & {1}\end{array}\right|=0 $$

Solve the system of linear equations. $$ \left\\{\begin{aligned} y-z+2 w =0 \\ 3 x+2 y +w=0 \\ 2 x +4 w=12 \\\\-2 x &-2 z+5 w= 6 \end{aligned}\right. $$

Nutrition A cat food manufacturer uses fish and beef byproducts. The fish contains 12 \(\mathrm{g}\) of protein and 3 \(\mathrm{g}\) of fat per ounce. The beef contains 6 \(\mathrm{g}\) of protein and 9 \(\mathrm{g}\) of \(\mathrm{fat}\) peounce. Fach can of cat food must contain at least 60 \(\mathrm{g}\) of protein and 45 \(\mathrm{g}\) of fat. Find a system of inequalities that describes the possible number of ounces of fish and beef that can be used in each can to satisfy these minimum requirements. Graph the solution set.

\(53-56\) Find \(x\) and \(y\) in terms of \(a\) and \(b\). $$ \left\\{\begin{aligned} a x+b y &=0 \\ x+y &=1 \end{aligned}(a \neq b)\right. $$

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (0,0),(6,2),(3,8) $$

Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$ \left\\{\begin{aligned} x+2 y &>4 \\\\-x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right. $$

\(53-56\) Find \(x\) and \(y\) in terms of \(a\) and \(b\). $$ \left\\{\begin{array}{l}{a x+b y=1} \\ {b x+a y=1}\end{array} \quad\left(a^{2}-b^{2} \neq 0\right)\right. $$

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (1,0),(3,5),(-2,2) $$

Mixtures A chemist has three acid solutions at various concentrations. The first is 10\(\%\) acid, the second is \(20 \%,\) and the third is \(40 \% .\) How many milliters of each should she use to make 100 \(\mathrm{mL}\) of 18\(\%\) solution, if she has to use four times as much of the 10\(\%\) solution as the 40\(\%\) solution?

\(53-56\) Find \(x\) and \(y\) in terms of \(a\) and \(b\). $$ \left\\{\begin{aligned} a x+b y &=0 \\ a^{2} x+b^{2} y &=1 \end{aligned} \quad(a \neq 0, b \neq 0, a \neq b)\right. $$

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (-1,3),(2,9),(5,-6) $$

Number Problem Find two numbers whose sum is 34 and whose difference is \(10 .\)