Exam 10: Systems of Equations and Inequalities

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

Find all solutions for the system. $\left\{ \begin{array} { l } y = x ^ { 2 } - 25 \\y - 5 = x\end{array} \right.$

Find the partial fraction decomposition of $\frac { 2 x ^ { 2 } + 3 x + 6 } { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 }$ .

A money manager invests $\$ 1,000,000$ of her client's money into three types of funds. It's estimated that Fund A will return $2 \%$ over the next year, Fund B $5 \%$ , and Fund C $11 \%$ . She decides to invest equal amounts in Fund A and Fund B and her client hopes to get a $\$ 45,500$ annual return. How much should she invest at each rate?

Find $A ^ { 2 }$ , or explain why the operation cannot be performed. $A = \left[ \begin{array} { l l l } 2 & 0 & 1 \\0 & 2 & 0 \\0 & 0 & 2\end{array} \right]$

The perimeter of a rectangle is $140$ in and its diagonal is $50$ in. Find its length and width.

A rectangular piece of sheet metal with an area of $1800$ in is bent into a cylindrical pipe having a volume of $900$ in . What are the dimensions of the sheet metal?

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

A right triangle has a perimeter of $20$ cm and an area of $10$ cm , what are the lengths of its sides.

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

Write out the form of the partial fraction decomposition of $\frac { 1 } { x ^ { 5 } - x ^ { -3 } }$ .

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + y + z = 2 \\2 x - 3 y + 2 z = 4 \\4 x + y - 3 z = 1\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } x + 2 y + z = 5 \\3 x + 7 y + 6 z = 29 \\2 x + 3 y = 0\end{array} \right.$

Solve the system. $\left\{ \begin{array} { c } 0.3 x - 0.1 y = 1.5 \\- 0.2 x + 0.4 y = - 3.0\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } u - 30 v = - 6 \\3 u - 80 v = - 6\end{array} \right.$

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. $\left\{ \begin{array} { l } x - y = 7 \\x + 2 y = 1\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } 2 x - y - 3 z = 3 \\x + 3 y - 2 z = - 4 \\7 x + 7 y - 12 z = - 5\end{array} \right.$

Use a calculator that can perform matrix operations to solve the system. $\left\{ \begin{array} { l } \frac { 1 } { 2 } x - \frac { 1 } { 3 } y - z = 5 \\x + 4 y + 2 z = 14 \\x + \frac { 3 } { 2 } y - 3 z = 13\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } 25 x - 75 y = 100 \\20 x - 60 y = 80\end{array} \right.$

Write the system of equations in matrix form. $\left\{ \begin{array} { l } 8 a - 3 b + c = 7 \\4 a - 1 b + c = 1 \\a + b + 2 c = 7 \\a = 0\end{array} \right.$