Exam 9: Systems and Matrices

Solve the system to find W₁ and W₂. -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 105 pounds is applied at the peak of the truss, then the forces or weights W1 and W2 Exerted parallel to each rafter of the truss are determined by the following linear system of Equations. $\frac { \sqrt { 3 } } { 2 } \left( W _ { 1 } + W _ { 2 } \right) = 105$ $W _ { 1 } - W _ { 2 } = 0$

Graph the inequality. - $x \leq 6$

Solve the problem. -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 95 black, 100 White and 85 red wires. How many of each type of cable were made?

Solve each problem. -Mommaʹs ice cream shop sells three types of ice cream: soft-serve, chunky, and nonfat. Location I sells 32 gal of soft-serve, 80 gal of chunky, and 30 gal of nonfat ice cream each day. Location II Sells 11 gal of soft-serve and Location III sells 60 gal of soft-serve each day. Daily sales of chunky Ice cream are 90 gal at Location II and 120 gal at Location III. At Location II, 30 gal of nonfat are Sold each day, and 40 gal of nonfat are sold each day at Location III. Write a 3 × 3 matrix that shows the sales figures for the three locations, with the rows representing The three locations. The incomes per gallon for soft-serve, chunky, and nonfat ice cream are $5, $4, And $6, respectively. Write a 3 × 1 matrix displaying the incomes. Find a matrix product that gives The daily income at each of the three locations.

Find the value of the determinant. - $\left| \begin{array} { r r r } 6 & 0 & 0 \\ 2 & - 4 & 0 \\ - 4 & 5 & 3 \end{array} \right|$

Solve the problem. -The sum of the squares of the digits of a positive two-digit number is 85, and the tens digit is 1 less than the units digit. Find the number.

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. - $P ( 1,5 ) , Q ( 5,7 ) , R ( 8,1 )$

Find the value of the determinant. - $\left| \begin{array} { c c } - 10 & - 2 \\0 & 9\end{array} \right|$

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 3 & 5 & 12 \\ 1 & 3 & 2 \\ 2 & - 3 & 3 \end{array} \right] ; - 2$ times row 2 added to row 3

Provide an appropriate response. -Suppose that $\mathrm { A }$ and $\mathrm { B }$ are both matrices of dimension $\mathrm { r } \times \mathrm { s }$ . Under what conditions can both the product $\mathrm { AB }$ and the product $\mathrm { BA }$ be found?

Solve the linear programming problem. -An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 900 tourist class, and 1500 economy class Passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first Class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and Can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of Each type of airplane should be used in order to minimize the operating cost?

Use Cramerʹs rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 2x-2y-6z=-22 4x+8y+5z=120 9x-3y+2z=65

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - +=90 -=72

Solve the equation for x. - $\left| \begin{array} { l l } 3 & x \\x & 4\end{array} \right| = - 4$

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

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. - $\left[ \begin{array} { r r r } - 8 & 3 & 3 \\ 6 & - 7 & - 6 \end{array} \right] \left[ \begin{array} { r r r } 7 & - 5 & 2 \\ - 3 & 1 & - 5 \\ - 5 & - 2 & 6 \end{array} \right]$

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. - $y = \ln ( 3 x - 2 )$ $x ^ { 2 } + 3 y ^ { 2 } = 7$

Provide an appropriate response. -2x - 7y = -17 5x + 3y = 19 If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

Write the augmented matrix for the system. Do not solve the system. - 6x-2y=-2 9y=-18