Chapter 10

Electricity Using Kirchhoff's Laws, it can be shown that the currents \(I_{1}, I_{2},\) and \(I_{3}\) that pass through the three branches of the circuit in the figure satisfy the given linear system. Solve the system to find \(I_{1}, I_{2},\) and \(I_{3} .\) $$ I_{1}+I_{2}-I_{3}=0 $$ $$16 I_{1}-8 I_{2} \quad=4$$ $$8 I_{2}+4 I_{3}=5$$

Find all solutions of the system of equations. \(\left\\{\begin{array}{l}{\frac{4}{x^{2}}+\frac{6}{y^{4}}=\frac{7}{2}} \\\ {\frac{1}{x^{2}}-\frac{2}{y^{4}}=0}\end{array}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{r}{x^{2}+y^{2} \leq 8} \\ {x \geqq 2} \\ {y \geq 0}\end{array}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{2 x^{3}+7 x+5}{\left(x^{2}+x+2\right)\left(x^{2}+1\right)}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} 2 x_{1}+3 x_{2}-5 x_{3} &=1 \\ x_{1}+x_{2}-x_{3} &=2 \\\ 2 x_{2}+x_{3} &=8 \end{aligned}\right. $$

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ B F+F E $$

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 correct to two decimal places, either by zooming in and using TRACE or by using Intersect. $$\left\\{\begin{array}{l}{2371 x-6552 y=13,591} \\ {9815 x+992 y=618,555}\end{array}\right.$$

35–46 Solve the system of linear equations. $$\left\\{\begin{array}{rr}{x+2 y-3 z=} & {-5} \\ {-2 x-4 y-6 z=} & {10} \\\ {3 x+7 y-2 z=} & {-13}\end{array}\right.$$

Agriculture A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It costs \(\$ 45\) per acre to grow corn, \(\$ 60\) for wheat, and \(\$ 50\) for soybeans. Because of market demand he will grow twice as many acres of wheat as of corn. He has allocated \(\$ 63,750\) for the cost of growing his crops. How many acres of each crop should he plant?

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=2 x+6} \\ {y=-x+5}\end{array}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{c}{x^{2}-y \geq 0} \\ {x+y<6} \\\ {x-y<6}\end{array}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{x^{2}+x+1}{2 x^{4}+3 x^{2}+1}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned}-2 a &+c=2 \\ a+2 b-c &=9 \\ 3 a+5 b+2 c &=22 \end{aligned}\right. $$

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ A B E $$

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 correct to two decimal places, either by zooming in and using TRACE or by using Intersect. $$\left\\{\begin{aligned}-435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right.$$

35–46 Solve the system of linear equations. $$\left\\{\begin{array}{l}{3 x-y+2 z=-1} \\ {4 x-2 y+z=-7} \\ {-x+3 y-2 z=-1}\end{array}\right.$$

Stock Portfolio An investor owns three stocks: \(A, B\) , and \(C\) . The closing prices of the stocks on three successive trading days are given in the table. $$ \begin{array}{|c|c|c|c|}\hline & {\text { Stock } A} & {\text { Stock B }} & {\text { Stock } C} \\ \hline \text { Monday } & {\$ 10} & {\$ 25} & {\$ 29} \\\ {\text { Tuesday }} & {\$ 12} & {\$ 20} & {\$ 32} \\ {\text { Wednesday }} & {\$ 16} & {\$ 15} & {\$ 32} \\ \hline\end{array} $$ Despite the volatility in the stock prices, the total value of the investor's stocks remained unchanged at \(\$ 74,000\) at the end of each of these three days. How many shares of each stock does the investor own?

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=-2 x+12} \\ {y=x+3}\end{array}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x^{2}+y^{2} &<9 \\ x+y &>0 \\ x & \leq 0 \end{aligned}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{x^{4}+x^{3}+x^{2}-x+1}{x\left(x^{2}+1\right)^{2}}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{5} y+\frac{1}{2} z &=\frac{7}{10} \\\\-\frac{2}{3} x+\frac{2}{5} y+\frac{3}{2} z &=\frac{11}{10} \\\ x-\frac{4}{5} y+z &=\frac{9}{5} \end{aligned}\right. $$

Solve for \(x\) and \(y\). $$ \left[\begin{array}{cc}{x} & {2 y} \\ {4} & {6}\end{array}\right]=\left[\begin{array}{cc}{2} & {-2} \\ {2 x} & {-6 y}\end{array}\right] $$

Find the inverse of the matrix. \(\left[\begin{array}{rr}{a} & {-a} \\ {a} & {a}\end{array}\right]\) \((a \neq 0)\)

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

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned}-x+2 y+z-3 w &=3 \\ 3 x-4 y+z+w &=9 \\\\-x-y+z+w &=0 \\\ 2 x+y+4 z-2 w &=3 \end{aligned}\right.$$

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

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=x^{2}+8 x} \\ {y=2 x+16}\end{array}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y & \geq x^{3} \\ y & \leq 2 x+4 \\ x+y & \geq 0 \end{aligned}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{2 x^{2}-x+8}{\left(x^{2}+4\right)^{2}}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} 2 x-y &=5 \\ 5 x &+3 z=19 \\ 4 y+7 z &=17 \end{aligned}\right. $$

Solve for \(x\) and \(y\). $$ 3\left[\begin{array}{ll}{x} & {y} \\ {y} & {x}\end{array}\right]=\left[\begin{array}{rr}{6} & {-9} \\ {-9} & {6}\end{array}\right] $$

Find the inverse of the matrix. \(\left[\begin{array}{cccc}{a} & {0} & {0} & {0} \\ {0} & {b} & {0} & {0} \\\ {0} & {0} & {c} & {0} \\ {0} & {0} & {0} & {d}\end{array}\right]\) \((a b c d \neq 0)\)

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

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned} x+y-z-w =6 \\ 2 x \qquad+z-3 w= 8 \\ x-y \qquad+4 w=-10 \\ 3 x+5 y-z-w =20 \end{aligned}\right.$$

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=x^{2}-4 x} \\ {2 x-y=2}\end{array}\right.\)

41–44 Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{array}{l}{y \geq x-3} \\ {y \geq-2 x+6} \\ {y \leq 8}\end{array}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{x^{5}-2 x^{4}+x^{3}+x+5}{x^{3}-2 x^{2}+x-2}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} 3 y+5 z &=4 \\ 2 x &-z=10 \\ 4 x+7 y &=0 \end{aligned}\right. $$

Solve for \(x\) and \(y\). $$ 2\left[\begin{array}{cc}{x} & {y} \\ {x+y} & {x-y}\end{array}\right]=\left[\begin{array}{rr}{2} & {-4} \\ {-2} & {6}\end{array}\right] $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{ll}{2} & {x} \\ {x} & {x^{2}}\end{array}\right]\)

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

35–46 Solve the system of linear equations. $$\left\\{\begin{array}{rrr}{x+y+2 z-} {w=} {-2} \\ {3 y+z+2 w} {=} {2} \\\ {x+y} \qquad {+3 w=} {2} \\ {-3 x} \qquad {+z+2 w=} {5}\end{array}\right.$$

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {x+3 y=2}\end{array}\right.\)

41–44 Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{aligned} x+y & \geq 12 \\ 2 x+y & \leq 24 \\ x-y & \geq-6 \end{aligned}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{x^{5}-3 x^{4}+3 x^{3}-4 x^{2}+4 x+12}{(x-2)^{2}\left(x^{2}+2\right)}\)

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{cc}{2 x-5 y} & {=4} \\ {x+y-z} & {=8} \\ {3 x} & {+5 z=0}\end{array}\right. $$

Solve for \(x\) and \(y\). $$ \left[\begin{array}{rr}{x} & {y} \\ {-y} & {x}\end{array}\right]-\left[\begin{array}{rr}{y} & {x} \\ {x} & {-y}\end{array}\right]=\left[\begin{array}{rr}{4} & {-4} \\ {-6} & {6}\end{array}\right] $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{cc}{e^{x}} & {-e^{2 x}} \\ {e^{2 x}} & {e^{3 x}}\end{array}\right]\)

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

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned} x-3 y+2 z+w =-2 \\ x-2 y \qquad -2 w=-10 \\ z+5 w =15 \\ 3 x \qquad +2 z+w=-3 \end{aligned}\right.$$