Exam 10: Systems of Equations and Inequalities

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } + 3 x - 5 } { ( x + 1 ) \left( x ^ { 2 } + 2 \right) }$ .

Solve the system of linear equations. $\left\{ \begin{array} { r } 2 x + 8 y - z = 2 \\- x - 3 y + z = 1 \\x + 4 y - z = 0\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y \leq 7 \\y > - 1 \\x \leq 2 \\y < 2 x - 3\end{array} \right.$

Find all solutions for the system. $\left\{ \begin{array} { l } \frac { 3 } { x } + \frac { 2 } { y } = 3 \\\\\frac { 2 } { x } - \frac { 5 } { y } = 1\end{array} \right.$

Find all the solutions for the system. $\left\{ \begin{array} { l } x - y = 5 \\x ^ { 2 } - y ^ { 2 } = 41\end{array} \right.$

Graph the inequality $y \leq 3 x - 1$ .

Find the inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? $\left[ \begin{array} { c c c } 2 & e ^ { x } & 0 \\e ^ { x } & - e ^ { 2 x } & 0 \\0 & 0 & 1\end{array} \right]$

Find the partial fraction decomposition of $\frac { 4 x ^ { 3 } + 1 } { \left( x ^ { 4 } + 2 x ^ { 2 } + 1 \right) }$ .

Find the partial fraction decomposition of $\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 4 } }$ .

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$ .

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

Solve the system. $\left\{ \begin{array} { l } 4 x - 12 y = 24 \\11 x - 4 y = 153\end{array} \right.$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $\left\{ \begin{array} { l } 4 x - y = 20 \\x - \frac { 1 } { 4 } y = 5\end{array} \right.$

Use a calculator that can perform matrix operations to solve the system. $\left\{ \begin{array} { l } x + 4 y + 7 z = 10 \\2 x + 5 y + 8 z = 11 \\3 x + 6 y - 9 z = 12\end{array} \right.$

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

Write out the form of the partial fraction decomposition of $\frac { 1 } { x ^ { 6 } - 1 }$ .

Given $A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]$ , solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

State whether the equation $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ is linear.

Use back-substitution to solve the triangular system. $\left\{ \begin{aligned}x - 3 y - z & = 0 \\2 y - 8 z & = 2 \\\frac { 1 } { 4 } z & = - 1\end{aligned} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } 3 x + y - 5 z = - 14 \\x - 2 z = - 4 \\- 7 x + 15 z = 31\end{array} \right.$