Exam 6: Systems and Matrices

A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 100 black, 110 white and 90 red Wires. How many of each type of cable were made?

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $( x + 3 ) ^ { 2 } + ( y - 3 ) ^ { 2 } \leq 4$

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - $\frac { 9 } { 7 } x + y = 5$ $0.4 \mathrm { x } - \mathrm { y } = 3$

A chair manufacturing company has two different factories. The profit function for Factory I when x chairs are manufactured and sold is as follows: $P ( x ) = - 0.05 x ^ { 2 } + 20 x - 500$ The profit function for Factory II when x chairs are manufactured and sold is as follows: $P ( x ) = - 0.05 x ^ { 2 } + 30 x - 2500$ For what number of chairs is the profit the same at both factories, and what is the profit?

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -If equations of the form y $y = f ( t )$ f(t) were determined that modeled either of the two graphs, then the variable t would represent _______ and the variable y would represent _______.

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - \pix+ey+z=2 ex+\piy+z=4 x+ey+\piz=6

Provide an appropriate response. -What is the value of $\left| \begin{array} { l l l } m & n & p \\0 & 0 & 0 \\q & r & s\end{array} \right|$ for any values of the variables?

Write the word or phrase that best completes each statement or answers the question. Which method should be used to solve the system? Explain your answer, including a description of the first step. -=1 6x+3y=-6

Suppose that you are solving a system of three linear equations by the Gauss-Jordan method and obtain the following augmented matrix. $\left[ \begin{array} { r r r | r } 1 & 7 & - 4 & 9 \\0 & 1 & 20 & 4 \\0 & 8 & 6 & - 9\end{array} \right]$ What row transformation would you perform next?

Solve the problem using matrices. -A chemist has prepared two acid solutions, one of which is $4 \%$ acid by volume, another $11 \%$ acid. How many cubic centimeters of each should the chemist mix together to obtain $50 \mathrm {~cm} ^ { 3 }$ of a solution that is $6.24 \%$ acid.

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. y=-14x+49 x+y=19

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \\ 3 \\ 2 \end{array} \right]$ . Find $C - 3 D$ .

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. - -4x-y-2z=-51 -6x+9z=-27 9y+z=84

A triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) has area equal to the absolute value of D, where $D = \frac { 1 } { 2 } \left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\x _ { 2 } & y _ { 2 } & 1 \\x _ { 3 } & y _ { 3 } & 1\end{array} \right|$ Find the area of the triangle having vertices at P, Q, and R. - $\mathrm { P } ( - 3,2 ) , \mathrm { Q } ( 1 , - 3 ) , \mathrm { R } ( 5,1 )$

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[ \begin{array} { r r r | r } 6 & - 2 & 8 & 8 \\ - 4 & - 4 & 0 & 4 \\ 8 & 0 & 2 & - 5 \end{array} \right]$

Solve the linear programming problem. -Zach is planning to invest up to $45,000 in corporate and municipal bonds. The least he will invest in corporate bonds is $7000 and he does not want to invest more than $27,000 in corporate bonds. He also does not want to Invest more than $34,262 in municipal bonds. The interest is 8.6% on corporate bonds and 6.2% on municipal Bonds. This is simple interest for one year. What is the maximum income?

Solve the system to find $W _ { 1 } \text { and } W _ { 2 }$ -Linear systems occur in the design of roof trusses for new homes and buildings. The simplest type of roof truss is a triangle. The truss shown in the figure is used to frame roofs of small buildings. If a force of 101 pounds is applied at the peak of the truss, then the forces or weights $W _ { 1 }$ and $W _ { 2 }$ exerted parallel to each rafter of the truss are determined by the following linear system of equations. $\frac { \sqrt { 3 } } { 2 } \left( W _ { 1 } + W _ { 2 } \right) = 101$ $W _ { 1 } - W _ { 2 } = 0$

The perimeter of a rectangle is 64 m. If the width were doubled and the length were increased by 5 m, the perimeter would be 100 m. What are the length and width of the rectangle?

Graph the solution set of the system of inequalities. - |x|\leq5 y\leq3 y\leq-3

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } - 2 & 4 \\4 & - 4\end{array} \right] \text { and } \left[ \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 4 } \\\frac { 1 } { 2 } & \frac { 1 } { 4 }\end{array} \right]$