Exam 10: Systems of Equations and Inequalities

Find the partial fraction decomposition of $\frac { x ^ { 4 } + 9 x ^ { 2 } + 25 + 2 x ^ { 3 } + 10 x } { x ^ { 5 } + 10 x ^ { 3 } + 25 x }$ .

Solve the system. $\left\{ \begin{array} { l } - \frac { 1 } { 5 } x + y = 4 \\- 4 x + 20 y = 80\end{array} \right.$

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }$ .

Given $A = \left[ \begin{array} { c c c } 6 & 2 & 0 \\10 & 0 & - 1\end{array} \right]$ , $B = \left[ \begin{array} { c c } 0 & - 3 \\1 & 0 \\- 1 & 3\end{array} \right]$ and $C = \left[ \begin{array} { c } 0 \\2 \\- 3\end{array} \right]$ , determine which of $A B C$ , $A C B$ , $B A C$ , $B C A$ , $C A B$ and $C B A$ are defined, and calculate them.

Write out the form of the partial fraction decomposition of $\frac { x ^ { -3 } - 3 x ^ { 2 } + x - 8 } { \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } - 2 x + 8 \right) }$ .

Use a determinant to find the area of the triangle shown.

There are 63 coins in Taylor's pocket, all in quarters and dimes. The total value is $9.60. Find the number of each type of coin.

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\x ^ { 2 } + y ^ { 2 } \leq 9\end{array} \right.$

Solve for $x$ . $\left| \begin{array} { c c c } ( x ) & 1 & 1 \\1 & x & x \\x & 1 & ( x + 1 )\end{array} \right| = 0$

Dana, Susan, and Caitlin enter a race in which they have to run, swim, and cycle over a marked course. Their average speeds are given in the table. Caitlin finishes first with a total time of 2 h 45 min. Dana comes in second with a time of 3 h 30 min. Susan finishes last with a time of 4 h 15 min. Find the distance (in miles) for each part of the race. Average speed (mi/h) Running Swimming Cycling Dana 10 4 20 Susan 5 4 20 Caitlin 10 3 40

Solve the system. $\left\{ \begin{array} { r } x + y = 8 \\2 x - 3 y = 1\end{array} \right.$

Find all solutions for the system. $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 18 \\x - y = 0\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } x - 7 z = - 41 \\- 3 x - 2 y + 7 z = 35 \\7 z = 49\end{array} \right.$

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { l } 5 x + 2 y - 3 z = 1 \\2 x + 5 y + z = - 4 \\- 3 x + 9 y - z = 5\end{array} \right.$

Write out the form of the partial fraction decomposition of $\frac { 1 } { ( x - 3 ) ( x + 2 ) }$ .

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + y + z = 10 \\2 x + 3 y + 2 z = 12 \\8 x - 4 y - 2 z = 16\end{array} \right.$

Find the inverse of the matrix $\left[ \begin{array} { l l } 1 & 0 \\2 & 1\end{array} \right]$ , if it exists.

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

Find the inverse of the matrix $\left[ \begin{array} { c c } 0 & - 1 \\- 3 & 2\end{array} \right]$ , if it exists.

State whether the system of equations is linear. $\left\{ \begin{array} { r } x + \sqrt { 2 } y + z = 2 \\3 x + 2 y - z = 0 \\x + y + z ^ { 2 } + 2 = 1\end{array} \right.$