Exam 7: Systems of Equations and Matrices

A bank gives three loans totaling $330,000 to a development company for the purchase of three business properties. The largest loan is $38,000 more than the sum of the other two. Find the amount of the largest loan.

Find the indicated sum or difference, if it is defined. - $\left[ \begin{array} { r } 5 x + 5 y \\- 6 k - 9 z \\2 w + 6 v \\- 2 m - 9 n\end{array} \right] - \left[ \begin{array} { c } 2 x + 2 y \\2 k - 4 z \\4 w + 9 v \\6 m - 3 n\end{array} \right]$

Find the inverse matrix of A. - $A = \left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$

Find the solution or solutions, if any exist, to the system. -A man has $\$ 298,000$ invested in three rental properties. One property earns $8.5 \%$ per year on the investment, a second earns $11 \%$ , and the third earns $7 \%$ . The annual earnings from the the properties total $\$ 17,000$ . Write a system of two equations to represent the problem with $\mathrm { x } , \mathrm { y }$ , and $\mathrm { z }$ representing the $8.5 \% , 11 \%$ , and $7 \%$ investments, respectively.

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } 4 x - 2 y = 26 \\4 x + 5 y = 33\end{array} \right.$

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } x + y = 1 \\x + y = - 7\end{array} \right.$

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$

A bank gives three loans totaling $280,000 to a development company for the purchase of three business properties. The largest loan is $40,000 more than the sum of the other two, and the smallest loan is one-third of The next larger loan. Find the amount of the largest loan.

Find the indicated sum or difference, if it is defined. -Compute the sum of $A = \left[ \begin{array} { r r } - 7 & 1 \\ 2 & 5 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 6 & 2 \\ 6 & 3 \end{array} \right]$ .

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } 2 x - 6 z - 6 w = - 4 \\x + y + z + 3 w = 2 \\4 x + 2 y - 4 z - 4 w = 0 \\6 x + 4 y - 2 z + 2 w = 4\end{array} \right.$

Find the indicated sum or difference, if it is defined. - $\left[ \begin{array} { l l } 5 & 4\end{array} \right] + \left[ \begin{array} { r } - 1 \\5\end{array} \right]$

The Hoffman Trucking Company has an order for three products, A, B, and C, for delivery. The table below gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of The products. If one of the company's trucks can carry 38,000 cubic feet and 57,100 pounds, and is insured to Carry $29,750, how many units of product A can be carried on the truck? Assume that z represents the number Of units of product C. Product A Product B Product C Unit Volume 24 20 50 Weight (lbs) 36 30 75 Value (dollars) 150 180 120

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } x - y + 3 z = - 8 \\ x + 5 y + z = 40 \\ 5 x + y + 13 z = 10 \end{array} \right.$

Solve the system graphically. -The supply function for a product is given by $p = q ^ { 2 } + 3 q + 125$ and the demand is given by $p = 555 - 30 q$ , where $\mathrm { p }$ is the price in dollars and $\mathrm { q }$ is the number of hundreds of units. Find the price that gives market equilibrium and the equilibrium quantity.

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { r } x - y + 3 z = - 8 \\x + 5 y + z = 40 \\5 x + y + 13 z = 10\end{array} \right.$

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12\end{array} \right.$

Find the value of the determinant. - $\left| \begin{array} { l l } 9 & 2 \\ 5 & 9 \end{array} \right|$

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & - 1 & 4 & - 20 \\5 & 0 & 1 & - 4 \\1 & 3 & 1 & 8\end{array} \right]$

Compute AB, if possible. - $A = \left[ \begin{array} { l l l } 4 & - 4 & - 9 \\1 & - 2 & - 6\end{array} \right] B = \left[ \begin{array} { r } 1 \\- 5 \\- 1\end{array} \right]$