Chapter 11

Write the system of equations as a matrix equation (see Example 6). $$ \left\\{\begin{array}{l}{2 x-5 y=7} \\ {3 x+2 y=4}\end{array}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} 5 x-3 y+z =6 \\ 4 y-6 z =22 \\ 7 x+10 y \quad=-13 \end{aligned}\right. $$

Solve the system of linear equations. $$ \left\\{\begin{aligned} 2 x-3 y+5 z &=14 \\ 4 x-y-2 z &=-17 \\\\-x-y+z &=3 \end{aligned}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ y & \leq 4 \\ 2 x+y & \leq 8 \end{aligned}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{array}{l}{y=e^{x}+e^{-x}} \\ {y=5-x^{2}}\end{array}\right. $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{aligned} 25 x-75 y &=100 \\\\-10 x+30 y &=-40 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x^{2}+x+1}{2 x^{4}+3 x^{2}+1} $$

Write the system of equations as a matrix equation (see Example 6). $$ \left\\{\begin{aligned} 6 x-y+z &=12 \\ 2 x\quad \quad+z &=7 \\ y-2 z &=4 \end{aligned}\right. $$

Solve the system of linear equations. $$ \left\\{\begin{array}{l}{2 x+y+3 z=9} \\ {-x \quad-7 z=10} \\ {3 x+2 y-z=4}\end{array}\right. $$

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

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{y>x+1} \\ {x+2 y \leq 12} \\\ {x+1>0}\end{array}\right. $$

Dimensions of a Rectangle A rectangle has an area of 180 \(\mathrm{cm}^{2}\) and a perimeter of \(54 \mathrm{cm} .\) What are its dimensions?

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

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{8 s-3 t=-3} \\ {5 s-2 t=-1}\end{array}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x^{4}+x^{3}+x^{2}-x+1}{x\left(x^{2}+1\right)^{2}} $$

Write the system of equations as a matrix equation (see Example 6). $$ \left\\{\begin{array}{r}{3 x_{1}+2 x_{2}-x_{3}+x_{4}=0} \\ {x_{1} \quad\quad\quad\quad-x_{3} \quad=5} \\ {3 x_{2}+x_{3}-x_{4}=4}\end{array}\right. $$

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

Solve the system of linear equations. $$ \left\\{\begin{aligned}-4 x-y+36 z &=24 \\ x-2 y+9 z &=3 \\\\-2 x+y+6 z &=6 \end{aligned}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x+y &>12 \\ y &<\frac{1}{2} x-6 \\ 3 x+y &<6 \end{aligned}\right. $$

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

Legs of a Right Triangle A right triangle has an area of 84 \(\mathrm{ft}^{2}\) and a hypotenuse 25 \(\mathrm{ft}\) long. What are the lengths of its other two sides?

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{aligned} u-30 v &=-5 \\\\-3 u+80 v &=5 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{2 x^{2}-x+8}{\left(x^{2}+4\right)^{2}} $$

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

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x^{2}+y^{2} & \leq 8 \\ x & \geq 2 \\ y & \geq 0 \end{aligned}\right. $$

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

Dimensions of a Rectangle The perimeter of a rectangle is \(70,\) and its diagonal is \(25 .\) Find its length and width.

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{\frac{1}{2} X+\frac{3}{5} y=3} \\ {\frac{5}{3} x+2 y=10}\end{array}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x^{5}-2 x^{4}+x^{3}+x+5}{x^{3}-2 x^{2}+x-2} $$

Let $$ \begin{array}{l}{A=\left[\begin{array}{llll}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right]} \\\ {B=\left[\begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right]}\end{array} $$ $$ C=\left[\begin{array}{r}{1} \\ {0} \\ {-1} \\ {-2}\end{array}\right] $$ Determine which of the following products are defined, and calculate the ones that are. $$ \begin{array}{cc}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array} $$

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

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x^{2}-y & \geq 0 \\ x+y &<6 \\ x-y &<6 \end{aligned}\right. $$

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

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x^{5}-3 x^{4}+3 x^{3}-4 x^{2}+4 x+12}{(x-2)^{2}\left(x^{2}+2\right)} $$

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 the system of linear equations. $$ \left\\{\begin{array}{l}{x-y+6 z=8} \\ {x \quad+\quad z=5} \\ {x+3 y-14 z=-4}\end{array}\right. $$

21-46 . 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 inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? $$ \left[\begin{array}{rrr}{1} & {e^{x}} & {0} \\ {e^{x}} & {-e^{2 x}} & {0} \\\ {0} & {0} & {2}\end{array}\right] $$

Flight of a Rocket A hill is inclined so that its "slope" is \(\frac{1}{2},\) as shown in the figure. We introduce a coordinate system with the origin at the base of the hill and with the scales on the axes measured in meters. A rocket is fired from the base of the hill in such a way that its trajectory is the parabola \(y=-x^{2}+401 x\) . At what point does the rocket strike the hillside? How far is this point from the base of the hill to the nearest centimeter)?

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{0.4 x+1.2 y=14} \\ {12 x-5 y=10}\end{array}\right. $$

Determine \(A\) and \(B\) in terms of \(a\) and \(b\) $$ \frac{a x+b}{x^{2}-1}=\frac{A}{x-1}+\frac{B}{x+1} $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} 2 x-5 y =4 \\ x+y-z =8 \\ 3 x +5 z=0 \end{aligned}\right. $$

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

21-46 . 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 inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? $$ \left[\begin{array}{cc}{x} & {1} \\ {-x} & {\frac{1}{x-1}}\end{array}\right] $$

Making a Stovepipe A rectangular piece of sheet metal with an area of 1200 \(\mathrm{in}^{2}\) is to be bent into a cylindrical length of stovepipe having a volume of \(600 \mathrm{in}^{3} .\) What are the dimensions of the sheet metal?

Determine \(A, B, C,\) and \(D\) in terms of \(a\) and \(b\) $$ \frac{a x^{3}+b x^{2}}{\left(x^{2}+1\right)^{2}}=\frac{A x+B}{x^{2}+1}+\frac{C x+D}{\left(x^{2}+1\right)^{2}} $$

Elecricity By 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 \(I_{3} .\) $$ \left\\{\begin{aligned} I_{1}+I_{2}-I_{3} &=0 \\ 16 I_{1}-8 I_{2} &=4 \\ 8 I_{2}+4 I_{3} &=5 \end{aligned}\right. $$

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