Exam 10: Systems of Equations and Inequalities

Perform the indicated operation, whenever possible. - $\left[ \begin{array} { r r } - 9 & - 9 \\- 8 & - 2 \\8 & 7\end{array} \right] + \left[ \begin{array} { l l } 5 & - 4 \\6 & - 2 \\2 & - 9\end{array} \right]$

Solve the problem. -The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 Man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the Company's profit, if the profit on a VIP ring is $40 and on an SST ring is $35?

Graph the system of inequalities. - + \leq36 + \geq4 A) no solution B) C) D)

Solve the problem. -A doctor has told a patient to take vitamin pills. The patient needs at least 28 units of vitamin A, at least 8 units of vitamin C, and at least 76 units of vitamin D. The red vitamin pills cost 20¢ each and contain 8 units of A, 1 Unit of C, and 7 units of D. The blue vitamin pills cost 40¢ each and contain 3 units of A, 1 unit of C, and 11 units Of D. How many pills should the patient take each day to minimize costs?

Solve the problem. -The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C daily. A cup of dietary drink X provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of Dietary drink Y provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Set up a system of linear Inequalities that describes the minimum daily requirements for calories and vitamins. Let x = number of cups of Dietary drink X, and y = number of cups of dietary drink Y. Write all the constraints as a system of linear Inequalities. A) $\left\{ \begin{array} { l } 60 x + 60 y \geq 300 \\ 12 x + 6 y > 36 \\ 10 x + 30 y \geq 90 \end{array} \right.$ B) $\left\{ \begin{array} { l } 60 x + 60 y \leq 300 \\ 12 x + 6 y \leq 36 \\ 10 x + 30 y \leq 90 \end{array} \right.$ C) $\left\{ \begin{array} { l } 60 x + 60 y > 3 \\ 12 x + 6 y > \\ 10 x + 30 y > \\ x > 0 \\ y > 0 \end{array} \right.$ D) $\left\{ \begin{array} { l } 60 x + 60 y \geq 300 \\ 12 x + 6 y \geq 36 \\ 10 x + 30 y \geq 90 \\ x \geq 0 \\ y \geq 0 \end{array} \right.$

Find the value of the determinant. - $\left| \begin{array} { r r r } 2 & 3 & 4 \\- 1 & - 4 & - 2 \\- 2 & - 4 & - 1\end{array} \right|$

Find the inverse of the matrix. - $\left[ \begin{array} { c c } - 4 & - 4 \\6 & 6\end{array} \right]$

Perform the matrix multiplication. -Let $A = \left[ \begin{array} { r r r } - 2 & - 9 & - 8 \\ 6 & - 9 & - 1 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 6 & - 4 & - 5 \\ 8 & - 4 & - 3 \\ - 3 & - 7 & 3 \end{array} \right]$ . Find $A B$ .

Write the word or phrase that best completes each statement or answers the question. -Find real numbers a, b, and c such that the graph of the function $y = a x ^ { 2 } + b x + c$ c contains the points (1, 1), (2, 4), and (-3, 29).

Write the partial fraction decomposition of the rational expression. - $\frac { 14 x + 1 } { ( x - 1 ) \left( x ^ { 2 } + x + 1 \right) }$

Use Cramer's rule to solve the linear system. - $\left\{ \begin{aligned}8 x - 3 z & = 29 \\2 x + 9 y + 5 z & = 104 \\2 x - 9 y & = - 31\end{aligned} \right.$

Write the partial fraction decomposition of the rational expression. - $\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }$

Solve the problem. -The difference of two numbers is 5 and the difference of their squares is 55. Find the numbers.

Graph the inequality. - $x>2$ A) B) C) D)

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 7 y = 7 \\3 x - 8 y = - 8\end{array} \right.$

Solve the problem. -The equation of the line passing through the distinct points $\left( x _ { 1 } , y _ { 1 } \right)$ and (x $\left. 2 , y _ { 2 } \right)$ is given by $\left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\ x _ { 2 } & y _ { 2 } & 1 \end{array} \right| = 0 .$ Find the equation of the line passing through the points $( 3,5 )$ and $( - 1,4 )$ .

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)? - +=52 2y+3x=0

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & 2 & 3\end{array} \right| = - 64 \left| \begin{array} { c c c } x - 2 & y - 4 & z - 6 \\- 2 u - 1 & - 2 v - 2 & - 2 w - 3 \\1 & 2 & 3\end{array} \right| = ?$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 4 \\x + y = 2\end{array} \right.$

Find the value of the determinant. - $\left| \begin{array} { r r r } - 2 & 5 & 4 \\3 & - 2 & 1 \\1 & 6 & - 3\end{array} \right|$