Chapter 6

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find $x, $$\begin{aligned} 5 x+y-3 z &=-6 \\ 2 x+3 y+z &=5 \\ -3 x-2 y+4 z &=3 \end{aligned}$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{rr|r} 2 & 1 & 1 \\\3 & -2 & -9 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}1 & 2 & 0 \\\\-1 & 2 & -1 \\\0 & 1 & 4\end{array}\right]$$

Your friend missed the lecture on adding matrices. In your own words, explain to her how to add two matrices.

Solve each system by substitution. $$\begin{aligned}&y=4 x-6\\\&2 x+5 y=-8\end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{6 x^{5}+7 x^{4}-x^{2}+2 x}{3 x^{2}+2 x-1}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.8 & -0.3 \\ 0.5 & -0.2 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$y>2(x+3)^{2}-1$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{rr|r} 1 & -5 & -18 \\ 6 & 2 & 20 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{ccc}2 & 1 & -1 \\\4 & 7 & -2 \\\2 & 4 & 0\end{array}\right]$$

Explain to a friend in your own words how to multiply a matrix by a scalar.

Solve each system by substitution. $$\begin{aligned}&3 x-2 y=12\\\&5 x=4-2 y\end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{x^{3}+4}{9 x^{3}-4 x}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$x^{2}+y^{2} \leq 4$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} x-3 y-2 z &=-3 \\ 3 x+2 y-z &=12 \\ -x-y+4 z &=3 \end{aligned}

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{ccc}10 & 2 & 1 \\\\-1 & 4 & 3 \\\\-3 & 8 & 10\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rrr} 6 & -9 & 2 \\ 4 & 1 & 3 \end{array}\right]+\left[\begin{array}{rrr} -8 & 2 & 5 \\ 6 & -3 & 4 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&8 x+3 y=2\\\&5 x=17+6 y\end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{x^{3}+2}{x^{3}-3 x^{2}+2 x}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$x^{2}+y^{2} \geq 1$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} x+y+z &=3 \\ 3 x-3 y-4 z &=-1 \\ x+y+3 z &=11 \end{aligned}

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|l} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}7 & -1 & 1 \\\1 & -7 & 2 \\\\-2 & 1 & 1\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr} 9 & 4 \\ -8 & 2 \end{array}\right]+\left[\begin{array}{rr} -3 & 2 \\ -4 & 7 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&4 x-5 y=-11\\\&2 x+y=5\end{aligned}$$

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

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 2 & 1 & 3 \\ -1 & 1 & 1 \end{array}\right]$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{array}{r} x-2 y+3 z=6 \\ 2 x-y+2 z=5 \end{array} \begin{aligned}

Graph each inequality. Do not use a calculator. $$x^{2}+(y+3)^{2} \leq 16$$

Write the system of equations associated with each augmented matrix. $$\begin{aligned} &(\mathrm{A})\\\ &\left[\begin{array}{cccc} 3 & 2 & 1 & 1 \\\ 0 & 2 & 4 & 22 \\ -1 & -2 & 3 & 15 \end{array}\right] \end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}1& -2 & 3 \\\0 & 0 & 0 \\\1 & 10 & -12\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr} -6 & 8 \\ 0 & 0 \end{array}\right]-\left[\begin{array}{rr} 0 & 0 \\ -4 & -2 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&7 x-y=-10\\\&3 y-x=10\end{aligned}$$

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

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} -2 & 1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right]$$

Write the system of equations associated with each augmented matrix. $$\begin{aligned} &(B)\\\ &\left[\begin{array}{cccc} 2 & 1 & 3 & 12 \\ 4 & -3 & 0 & 10 \\ 5 & 0 & -4 & -11 \end{array}\right] \end{aligned}$$

Graph each inequality. Do not use a calculator. $$(x-4)^{2}+y^{2} \leq 9$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & 9 & 0 \\\\-1 & -2 & 0\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr} 1 & -4 \\ 2 & -3 \\ -8 & 4 \end{array}\right]-\left[\begin{array}{rr} -6 & 9 \\ -2 & 5 \\ -7 & -12 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&4 x+5 y=7\\\&9 y=31+2 x\end{aligned}$$

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} 3 x+4 y-z &=13 \\ x+y+2 z &=15 \end{aligned}

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & 2 & 3 \\ 0 & 1 & -1 \end{array}\right]$$

In your own words, explain how to determine whether the boundary of the graph of an inequality is solid or dashed.

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}3 & 3 & -1 \\\2 & 6 & 0 \\\\-6 & -6 & 2\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr} 6 & -2 \\ 5 & 4 \end{array}\right]+\left[\begin{array}{rr} -1 & 7 \\ 7 & -4 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&2 x+6 y=-18\\\&5 y=-29+3 x\end{aligned}$$