Exam 10: Systems of Equations and Inequalities

Graph the system of inequalities. - $\left\{ \begin{aligned}2 x + 3 y & \geq 6 \\x - y & \geq 3 \\y & \leq 2\end{aligned} \right.$ A) B) C) D)

Solve the system of equations by using substitution. - $\left\{ \begin{array} { l } 5 x + 3 y = 80 \\2 x + y = 30\end{array} \right.$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } y = - x ^ { 2 } + 3 \\x ^ { 2 } + y ^ { 2 } = 5\end{array} \right.$

Write the augmented matrix for the system. - $\left\{ \begin{aligned}4 x + 9 y + 2 z & = 30 \\- 2 x + 6 y + 2 z & = 0 \\9 x + 6 y + 5 z & = 38\end{aligned} \right.$

Set up the linear programming problem. -The Jillson's have up to $75,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 6% and that no more than $20,000 should be invested in more volatile bonds yielding 12%. (a) Using x to denote the amount of money invested in the stable bonds and y the amount invested in the more volatile bonds, write a system of linear inequalities that describe the possible amounts of each investment.(b) Graph the system and label the corner points.

Perform the indicated matrix operations. - $\text { Let } C = \left[ \begin{array} { r } 1 \\- 3 \\2\end{array} \right] \text { and } D = \left[ \begin{array} { r } - 1 \\3 \\- 2\end{array} \right] \text {. Find } C - 3 D \text {. }$ A) $\left[ \begin{array} { c } - 4 \\ 12 \\ - 8 \end{array} \right]$ B) $\left[ \begin{array} { r } 4 \\ - 6 \\ 4 \end{array} \right]$ C) $\left[ \begin{array} { r } - 2 \\ 6 \\ - 4 \end{array} \right]$ D) $\left[ \begin{array} { r } 4 \\ - 12 \\ 8 \end{array} \right]$

Write the partial fraction decomposition of the rational expression. - $\frac { 2 x ^ { 3 } - 2 x ^ { 2 } + 15 x - 9 } { \left( x ^ { 2 } + 5 \right) ^ { 3 } }$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x ^ { 2 } - y ^ { 2 } = 39 \\x - y = 3\end{array} \right.$

Solve the system of equations by using substitution. - $\left\{ \begin{array} { r } x + 7 y = - 2 \\3 x + y = 34\end{array} \right.$

Solve the system. - $\left\{ \begin{array} { l } x - 4 y = - 10 \\2 x - 8 y = - 17\end{array} \right.$

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

Perform the matrix multiplication. -Let $A = \left[ \begin{array} { r r } - 2 & 3 \\ 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 2 & 0 \\ - 1 & 3 \end{array} \right]$ . Find $A B$ .

Find the inverse of the matrix. - $\left[ \begin{array} { r r } - 1 & 6 \\0 & 2\end{array} \right]$

Find the value(s) of the function, subject to the system of inequalities. -Find the maximum and minimum of $z = 19 x + 7 y \text { subject to } x \geq 0 , y \geq 0,2 x + 3 y \geq 6 , x \leq 10 , y \leq 5$

Perform the indicated matrix operations. -Let $A = \left[ \begin{array} { l l } 1 & 3 \\ 2 & 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ Find $2 A + B$ A) $\left[ \begin{array} { l l } 2 & 10 \\ 3 & 18 \end{array} \right]$ B) $\left[ \begin{array} { l l } 2 & 10 \\ 1 & 12 \end{array} \right]$ C) $\left[ \begin{array} { l l } 2 & 14 \\ 2 & 24 \end{array} \right]$ D) $\left[ \begin{array} { r r } 2 & 7 \\ 3 & 12 \end{array} \right]$

Write the partial fraction decomposition of the rational expression. - $\frac { 12 x ^ { 2 } + 162 x + 384 } { ( x + 8 ) ( x + 2 ) ( x + 11 ) }$

Graph the inequality. - $y > x ^ { 2 } + 2$ A) B) C) D)

Find the inverse of the matrix. - $\left[ \begin{array} { r r } 0 & - 3 \\- 3 & - 6\end{array} \right]$

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & 1 & 1\end{array} \right| = - 6 \quad \left| \begin{array} { l l l } u & v & w \\2 & 2 & 2 \\x & y & z\end{array} \right| = ?$

The graph of two equations along with the points of intersection are given. Substitute the points of intersection into the systems of equations. Are the points of intersection solutions to the system of equations (Y/N)? - +=25 2y+x=5