Chapter 11

For Problems \(11-28\), solve each system by using the substitution method. \(\left(\begin{array}{l}x+y=16 \\ y=x+2\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-3 x+1}{(x+1)^{2}}\)

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. \(\left(\begin{array}{l}6 x-5 y=1 \\ 4 x-7 y=2\end{array}\right)\)

Use a matrix approach to solve each system. \(\left(\begin{array}{r}x+5 y=-18 \\ -2 x+3 y=-16\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}x=3 y-25 \\ 4 x+5 y=19\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-6 x^{2}+19 x+21}{x^{2}(x+3)}\)

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. \(\left(\begin{array}{c}-\frac{2}{3} x+\frac{1}{2} y=-7 \\ \frac{1}{3} x-\frac{3}{2} y=6\end{array}\right)\)

For Problems 13-28, evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}1 & 2 & -1 \\ 3 & 1 & 2 \\ 2 & 4 & 3\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}3 x-4 y & =33 \\ x+7 y & =-39\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}x=3 y-25 \\ 4 x+5 y=19\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{10 x^{2}-73 x+144}{x(x-4)^{2}}\)

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. \(\left(\begin{array}{l}\frac{1}{2} x+\frac{2}{3} y=-6 \\ \frac{1}{4} x-\frac{1}{3} y=-1\end{array}\right)\)

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

Use a matrix approach to solve each system. \(\left(\begin{array}{c}2 x+7 y=-55 \\ x-4 y=25\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}3 x-5 y=25 \\ x=y+7\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{-2 x^{2}-3 x+10}{\left(x^{2}+1\right)(x-4)}\)

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

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

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x-6 y & =-2 \\ 2 x-12 y & =5\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}y=\frac{2}{3} x-1 \\ 5 x-7 y=9\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{8 x^{2}+15 x+12}{\left(x^{2}+4\right)(3 x-4)}\)

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

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

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

Solve each system by using the substitution method. \(\left(\begin{array}{l}y=\frac{3}{4} x+5 \\ 4 x-3 y=-1\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{3 x^{2}+10 x+9}{(x+2)^{3}}\)

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. \(\left(\begin{array}{rl}x-y+2 z & =-8 \\ 2 x+3 y-4 z & =18 \\ -x+2 y-z & =7\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}6 & 12 & 3 \\ -1 & 5 & 1 \\ -3 & 6 & 2\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{l}3 x-5 y=39 \\ 2 x+7 y=-67\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}a=4 b+13 \\ 3 a+6 b=-33\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{2 x^{3}+8 x^{2}+2 x+4}{(x+1)^{2}\left(x^{2}+3\right)}\)

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

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & 35 & 5 \\ 1 & -5 & 1 \\ -4 & 15 & 2\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}3 x+9 y & =-1 \\ x+3 y & =10\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{c}9 a-2 b=28 \\ b=-3 a+1\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{5 x^{2}+3 x+6}{x\left(x^{2}-x+3\right)}\)

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. \(\left(\begin{array}{c}2 x-3 y+z=-7 \\ -3 x+y-z=-7 \\ x-2 y-5 z=-45\end{array}\right)\)

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

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

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

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

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. \(\left(\begin{array}{rl}3 x-y-z & =18 \\ 4 x+3 y-2 z & =10 \\ -5 x-2 y+3 z & =-22\end{array}\right)\)

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & -17 & 3 \\ 0 & 5 & 1 \\ 1 & -3 & -1\end{array}\right|\)

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x+3 y-4 z & =13 \\ 2 x+7 y-3 z & =11 \\ -2 x-y+2 z & =-8\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}t+u=11 \\ t=u+7\end{array}\right)\)

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{2 x^{3}+x+3}{\left(x^{2}+1\right)^{2}}\)

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. \(\left(\begin{array}{c}4 x+5 y-2 z=-14 \\ 7 x-y+2 z=42 \\ 3 x+y+4 z=28\end{array}\right)\)

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

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}-2 x-5 y+3 z & =11 \\ x+3 y-3 z & =-12 \\ 3 x-2 y+5 z & =31\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}u=t-2 \\ t+u=12\end{array}\right)\)