Exam 6: Systems and Matrices

Perform the operation or operations when possible. - $\left[ \begin{array} { r r } - 7 & - 7 \\4 & 8\end{array} \right] + \left[ \begin{array} { r r } 1 & - 4 \\- 4 & - 9\end{array} \right] - \left[ \begin{array} { r r } 6 & 9 \\- 4 & - 4\end{array} \right]$

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { r r r } a & 5 & 3 \\ - 7 & 3 & f \end{array} \right] + \left[ \begin{array} { r r r } - 3 & b & 8 \\ d & - 4 & - 6 \end{array} \right] = \left[ \begin{array} { r r r } 6 & - 3 & c \\ 6 & e & 6 \end{array} \right]$

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Find the partial fraction decomposition for the rational expression. $\frac { 63 x ^ { 2 } - 7 x ^ { 3 } } { x ^ { 2 } ( 9 - x ) }$

Solve the equation for x. $\left| \begin{array} { r r } - 2 & 5 \\1 & x\end{array} \right| = - 9$

Solve the linear programming 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?

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

Find the partial fraction decomposition for the rational expression. - $\frac { 4 x ^ { 2 } + 3 x + 56 } { \left( x ^ { 2 } - 64 \right) ( x - 1 ) }$

For any augmented matrix of a system of linear equations, which of the following row transformations will not necessarily result in a matrix of an equivalent system? (i) Adding to the elements of any row a multiple of the corresponding elements of another row. (ii) Interchanging any two columns. (iii) Multiplying the elements of any row by the corresponding elements of another row. (iv) Multiplying the elements of any row by any nonzero number.

Solve the linear programming problem. -An automotive glass plant must be able to fill orders for 230 truck windshields, 350 auto windshields, and 310 RV windshields. Because the first-shift crews are more experienced, they are each able to make 7 truck Windshields, 12 auto windshields, and 9 RV windshields. The second-shift crews are able to produce 4 truck Windshields, 12 auto windshields, and 7 RV windshields. The first-shift crews cost the company $360.00 per Crew per shift, and the second-shift crews cost $310.00 per crew per shift. The company's facilities will handle at Most 50 crews on each shift, and they must have at least 10 crews come in to keep the plant running. How many First-shift and how many second-shift crews should be scheduled to minimize costs?

Find the partial fraction decomposition for the rational expression. $\frac { - 3 x ^ { 2 } - 5 x - 42 } { ( x - 3 ) \left( x ^ { 2 } + 3 \right) }$

Graph the solution set of the system of inequalities. - y\leq--6x-4 y\geq+6x+4

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - +=13.85 x=3y-1

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { r r r } 9 & 8 & - 6 \\1 & \mathrm {~m} & - 2\end{array} \right] = \left[ \begin{array} { l l l } \mathrm { x } & \mathrm { y } & - 6 \\1 & 3 & - 2\end{array} \right]$

Graph the solution set of the system of inequalities. - y-x\leq5 x+y\geq3 y-3x\geq-1

Find the dimensions of a rectangular enclosure with perimeter 34 ft and area 6 $60 \mathrm { ft } ^ { 2 }$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. 2+xy+2=2 3-5xy+3=3

Find the matrix product when possible. -Given $A = \left[ \begin{array} { r r } 2 & - 2 \\ - 4 & - 5 \end{array} \right]$ , find $A ^ { 2 }$ .

Which method should be used to solve the system? Explain your answer, including a description of the first step. - -3 =49 7-7 =9

Provide an appropriate response. -Fill in the blank to complete the statement. A matrix having the same number of rows as columns is called a _______ matrix.