Chapter 11

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{4 x^{2}+3 x+14}{x^{3}-8}\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}-5 x+6 y+4 z & =-4 \\ -7 x-8 y+2 z & =-2 \\ 2 x+9 y-z & =1\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}-5 & 1 & -1 \\ 3 & 4 & 2 \\ 0 & 2 & -3\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}-3 x+2 y+z & =17 \\ x-y+5 z & =-2 \\ 4 x-5 y-3 z & =-36\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}y=5 x-9 \\ 5 x-y=9\end{array}\right)\)

Give a general description of partial fraction decomposition for someone who missed class the day it was discussed.

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}2 x-y+3 z=-17 \\ 3 y+z=5 \\ x-2 y-z=-3\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}3 & -4 & -2 \\ 5 & -2 & 1 \\ 1 & 0 & 0\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x-3 y-z & =2 \\ 3 x+y-4 z & =-18 \\ -2 x+5 y+3 z & =2\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x+3 y=-7 \\ 3 x-2 y=16\end{array}\right)\)

Give a step-by-step explanation of how to find the partial fraction decomposition of \(\frac{11 x+5}{2 x^{2}+5 x-3}\).

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}2 x-y+3 z=-5 \\ 3 x+4 y-2 z=-25 \\\ -x+z=6\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}-6 & 5 & 3 \\ 2 & 0 & -1 \\ 4 & 0 & 7\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x-4 y+3 z & =16 \\ 2 x+3 y-4 z & =-22 \\ -3 x+11 y-z & =-36\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}5 x-3 y=-34 \\ 2 x+7 y=-30\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}x+3 y-4 z=-1 \\ 2 x-y+z=2 \\ 4 x+5 y-7 z=0\end{array}\right)\)

Use a matrix approach to solve each system. \(\left(\begin{array}{r}x-y+2 z=1 \\ -3 x+4 y-z=4 \\ -x+2 y+3 z=6\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}x-2 y+z=1 \\ 3 x+y-z=2 \\ 2 x-4 y+2 z=-1\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & -1 & 3 \\ 0 & 3 & 1 \\ 4 & -8 & -4\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x+2 y-5 z & =-1 \\ 2 x+3 y-2 z & =2 \\ 3 x+5 y-7 z & =4\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}2 x+3 y=3 \\ 4 x-9 y=-4\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}3 x-2 y-3 z & =-5 \\ x+2 y+3 z & =-3 \\ -x+4 y-6 z & =8\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & 3 & -4 \\ 4 & 6 & -1 \\ -6 & 1 & -2\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}-2 x+y+5 z & =-5 \\ 3 x+8 y-z & =-34 \\ x+2 y+z & =-12\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x-5 y=3 \\ 8 x+15 y=-24\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}3 x-2 y+z & =11 \\ 5 x+3 y & =17 \\ x+y-2 z & =6\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}1 & 2 & -3 \\ -3 & -1 & 1 \\ 4 & 5 & 4\end{array}\right|\)

Use a matrix approach to solve each system. \(\left.\begin{array}{rl}4 x-10 y+3 z & =-19 \\ 2 x+5 y-z & =-7 \\ x-3 y-2 z & =-2\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{c}4 x+y=9 \\ y=15-4 x\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}x-2 y+3 z & =1 \\ -2 x+4 y-3 z & =-3 \\ 5 x-6 y+6 z & =10\end{array}\right)\)

For Problems \(29-32\), evaluate each \(4 \times 4\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrrr}1 & -2 & 3 & 2 \\ 2 & -1 & 0 & 4 \\ -3 & 4 & 0 & -2 \\ -1 & 1 & 1 & 5\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}2 x+3 y-z & =7 \\ 3 x+4 y+5 z & =-2 \\ 5 x+y+3 z & =13\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}3 x+2 y=1 \\ 5 x-2 y=23\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}2 x-y+2 z=-1 \\ 4 x+3 y-4 z=2 \\ x+5 y-z=9\end{array}\right)\)

Evaluate each \(4 \times 4\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrrr}1 & 2 & 5 & 7 \\ -6 & 3 & 0 & 9 \\ -3 & 5 & 2 & 7 \\\ 2 & 1 & 4 & 3\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{r}4 x+3 y-z=0 \\ 3 x+2 y+5 z=6 \\ 5 x-y-3 z=3\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x+3 y=-22 \\ 4 x-5 y=26\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}-x-y+3 z & =-2 \\ -2 x+y+7 z & =14 \\ 3 x+4 y-5 z & =12\end{array}\right)\)

Evaluate each \(4 \times 4\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrrr}3 & -1 & 2 & 3 \\ 1 & 0 & 2 & 1 \\ 2 & 3 & 0 & 1 \\\ 5 & 2 & 4 & -5\end{array}\right|\)

Subscript notation is frequently used for working with larger systems of equations. For Problems 31-34, use a matrix approach to solve each system. Express the solutions as 4-tuples of the form \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\).\(\left(\begin{array}{rl}x_{1}-3 x_{2}-2 x_{3}+x_{4} & =-3 \\ -2 x_{1}+7 x_{2}+x_{3}-2 x_{4} & =-1 \\ 3 x_{1}-7 x_{2}-3 x_{3}+3 x_{4} & =-5 \\\ 5 x_{1}+x_{2}+4 x_{3}-2 x_{4} & =18\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{rl}x-3 y & =-22 \\ 2 x+7 y & =60\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{c}-2 x+y-3 z=-4 \\ x+5 y-4 z=13 \\ 7 x-2 y-z=37\end{array}\right)\)

Evaluate each \(4 \times 4\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrrr}1 & 2 & 0 & 0 \\ 3 & -1 & 4 & 5 \\ -2 & 4 & 1 & 6 \\\ 2 & -1 & -2 & -3\end{array}\right|\)

Subscript notation is frequently used for working with larger systems of equations. For Problems 31-34, use a matrix approach to solve each system. Express the solutions as 4-tuples of the form \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\).\(\left(\begin{array}{rl}x_{1}-2 x_{2}+2 x_{3}-x_{4} & =-2 \\ -3 x_{1}+5 x_{2}-x_{3}-3 x_{4} & =2 \\ 2 x_{1}+3 x_{2}+3 x_{3}+5 x_{4} & =-9 \\\ 4 x_{1}-x_{2}-x_{3}-2 x_{4} & =8\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{rl}6 x-y & =3 \\ 5 x+3 y & =-9\end{array}\right)\)

Give a step-by-step description of how you would solve the system $$ \left(\begin{array}{rl} 2 x-y+3 z & =31 \\ x-2 y-z & =8 \\ 3 x+5 y+8 z & =35 \end{array}\right) $$

Subscript notation is frequently used for working with larger systems of equations. For Problems 31-34, use a matrix approach to solve each system. Express the solutions as 4-tuples of the form \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\).\(\left(\begin{array}{rl}x_{1}+3 x_{2}-x_{3}+2 x_{4} & =-2 \\ 2 x_{1}+7 x_{2}+2 x_{3}-x_{4} & =19 \\ -3 x_{1}-8 x_{2}+3 x_{3}+x_{4} & =-7 \\\ 4 x_{1}+11 x_{2}-2 x_{3}-3 x_{4} & =19\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x-5 y=21 \\ 3 x+7 y=-38\end{array}\right)\)

Give a step-by-step description of how you would finc the value of \(x\) in the solution for the system $$ \left(\begin{array}{rl} x+5 y-z & =-9 \\ 2 x-y+z & =11 \\ -3 x-2 y+4 z & =20 \end{array}\right) $$

Use the appropriate property of determinants from this section to justify each true statement. Do not evaluate the determinants. \(\left|\begin{array}{rrr}1 & -2 & 3 \\ 4 & -6 & -8 \\ 0 & 2 & 7\end{array}\right|=(-2)\left|\begin{array}{rrr}1 & -2 & 3 \\ -2 & 3 & 4 \\\ 0 & 2 & 7\end{array}\right|\)