Exam 10: Systems of Equations and Inequalities

Use a graphing device to find all solutions of the system. $\left\{ \begin{array} { l } x ^ { 2 } - 2 x = y ^ { 2 } \\x - y ^ { 5 } = 0\end{array} \right.$

Find the inverse of the matrix $\left[ \begin{array} { c c c } - 2 & 1 & - 1 \\0 & 2 & 0 \\4 & 3 & 3\end{array} \right]$ , if it exists.

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

Evaluate the minor $M _ { 33 }$ and the cofactor $A _ { 33 }$ using the matrix $A = \left[ \begin{array} { c c c } 0 & \frac { 1 } { 4 } & 8 \\\frac { - 1 } { 2 } & - 7 & - 3 \\5 & 4 & 22\end{array} \right]$ .

Solve the system of linear equations. $\left\{ \begin{array} { r } x + 3 y = 0 \\2 x + 7 y + 2 z = - 1 \\x + 4 y + 3 z = - 4\end{array} \right.$

Graph the inequality $y < 5$ .

Use a graphing device to find all solutions of the system. $\left\{ \begin{array} { l } y = 2 x ^ { 2 } + 3 \\y = - 2 x + 5\end{array} \right.$

Graph the inequality $x \geq 3$ .

Solve the system. $\left\{ \begin{array} { l } 8 x - 3 y = 5 \\5 x - 2 y = 4\end{array} \right.$

Given $A = \left[ \begin{array} { c c c } 2 & 3 & - 1 \\4 & - 2 & 0 \\- 3 & - 5 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 3 & 5 & - 3 \\5 & 2 & 2 \\- 7 & 4 & 2\end{array} \right]$ , find $3 A - 2 B$ , or explain why the operation cannot be performed.

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y \leq 3 \\x > - 2 \\3 x - 2 y < 5\end{array} \right.$

Evaluate the minor $M _ { 32 }$ and the cofactor $A _ { 32 }$ using the matrix $A = \left[ \begin{array} { c c c } 7 & 0 & 9 \\1 & - 8 & - 3 \\5 & 4 & 6\end{array} \right]$ .

Find all solutions for the system. $\left\{ \begin{array} { l } y + x ^ { 2 } = 4 x \\y + 2 x = 8\end{array} \right.$

Find the partial fraction decomposition of $\frac { x ^ { 2 } - 3 } { ( - 3 x + 5 ) \left( x ^ { 2 } - x + 6 \right) }$ .

Solve for $x$ and $y$ . $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$

Find all solutions for the system. $\left\{ \begin{array} { l } 5 ^ { x } + 5 ^ {y } = 10 \\25 ^ { x } - 25^ {y } = 75\end{array} \right.$ Note that $25 ^ { x } = 5 ^ { 2 x } = \left( 5 ^ { x } \right) ^ { 2 }$ .

Find the inverse of the matrix $\left[ \begin{array} { c c } - 2 & - 1 \\4 & 3\end{array} \right]$ , if it exists.

Solve the system of linear equations. $\left\{ \begin{array} { r } 2 x + 3 y = 11 \\3 x + 3 y = 9 \\x + z = 1\end{array} \right.$

Graph the inequality $x - 3 y \geq 4$ .

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