Exam 10: Systems of Equations and Inequalities

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } - x + 3 y - 2 z = - 1 \\4 x - 2 y + z = - 7 \\3 x - y + 2 z = - 1\end{array} \right.$

Find the inverse of the matrix, if it exists. $\left[ \begin{array} { l l l l } 1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\0 & 0 & 1 & 1\end{array} \right]$

Given $A = \left[ \begin{array} { c c } 3 & 1 \\- 2 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } 0 & 3 \\1 & - 5\end{array} \right]$ , find $A + B$ , or explain why the operation cannot be performed.

Find the inverse of the matrix $\left[ \begin{array} { c c } 2 & 1 \\- 4 & - 2\end{array} \right]$ , if it exists.

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } 2 x + 2 y - 2 z = 0 \\x + y + 2 z = - 1 \\x + y - z = 2\end{array} \right.$

Find the inverse of the matrix, if it exists. $\left[ \begin{array} { r r } \frac { 1 } { 2 } & 4 \\8 & - 1\end{array} \right]$

Find the partial fraction decomposition of $\frac { x ^ { 4 } - 4 x ^ { 3 } - 13 x ^ { 2 } + 49 x - 9 } { x ^ { 3 } - 4 x ^ { 2 } - 9 x + 36 }$ .

The sum of the digits of a two digit number is 10. When the digits are interchanged, the number is decreased by $36$ . Find the number.

Find the determinant of the matrix, if it exists. $\left[ \begin{array} { l l } \frac { 1 } { 8 } & \frac { 1 } { 4 } \\\\\frac { 3 } { 2 } & 1\end{array} \right]$

Determine whether the matrix $\left[ \begin{array} { l l l l } 1 & 0 & 0 & 2 \\0 & 1 & 0 & 3 \\0 & 0 & 1 & 4\end{array} \right]$ is in echelon or reduced echelon form. Write the corresponding system of equations.

Find the partial fraction decomposition of $\frac { x + 15 } { x ^ { 2 } + 3 x }$ .

Determine whether the matrix $\left[ \begin{array} { l l l } 1 & 2 & 0 \\0 & 1 & 3\end{array} \right]$ is in echelon or reduced echelon form. Write the corresponding system of equations.

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } 4 x + 5 y = 0 \\2 x + 2 y + 2 z = - 1 \\x + 4 y + 3 z = 2\end{array} \right.$

Given $A = \left[ \begin{array} { c c } 1 & 5 \\- 2 & 3\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]$ , find $B A$ , or explain why the operation cannot be performed.

A boat travels upstream (against the current) between two points on a river $10$ mi apart in $2.5$ hours. The return trip downstream takes only $45$ minutes. What is the boat's speed, and how fast does the current in the river flow?

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

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + 2 y - z = - 2 \\2 x - 3 y + 5 z = - 4 \\4 x + 8 y - 4 z = - 8\end{array} \right.$

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + 2 y - 3 z = - 5 \\3 x + 7 y + 2 z = - 13 \\- 2 x - 4 y - 6 z = 10\end{array} \right.$

Use the graphical method to find all the solutions of the system of equations. $\left\{ \begin{array} { l } y = 1 - 8 x ^ { 3 } \\y = 4 x ^ { 3 } - 2 x ^ { 2 }\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y > \frac { 1 } { 4 } x \\y \leq 4 x + 2\end{array} \right.$