91Ó°ÊÓ

Graph each system of equations as a pair of lines in the \(x y\) -plane. Solve each system and interpret your answer. $$\begin{array}{l} 3 x-5 y=7 \\ 2 x+y=9 \end{array}$$

Find the solution set of the system of linear equations represented by the augmented matrix. $$\left[\begin{array}{ccccc} 1 & 2 & 0 & 1 & 4 \\ 0 & 1 & 2 & 1 & 3 \\ 0 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 1 & 4 \end{array}\right]$$

Complete the following set of tasks for each system of equations. (a) Use a graphing utility to graph the equations in the system. (b) Use the graphs to determine whether the system is consistent or inconsistent. (c) If the system is consistent, approximate the solution. (d) Solve the system algebraically. (e) Compare the solution in part (d) with the approximation in part (c). What can you conclude? $$\begin{array}{r} 4 x-5 y=3 \\ -8 x+10 y=14 \end{array}$$

Find the values of \(x, y,\) and \(\lambda\) that satisfy the system of equations. Such systems arise in certain problems of calculus, and \(\lambda\) is called the Lagrange multiplier. $$\begin{array}{rr} 2 x+\lambda & =0 \\ 2 y+\lambda & =0 \\ x+y & -4=0 \end{array}$$

In the 2007 Fiesta Bowl Championship Series on January 8 \(2007,\) the University of Florida Gators defeated the Ohio State University Buckeyes by a score of 41 to \(14 .\) The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth \(6,1,\) and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. The number of touchdowns was one more than three times the number of field goals. Write a system of equations to represent this event. Then determine the number of each type of scoring play. (Source: www.fiestabowl.org)

Consider the matrix \(A=\left[\begin{array}{rrr}1 & k & 2 \\ -3 & 4 & 1\end{array}\right]\) (a) If \(A\) is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. (b) If \(A\) is the augmented matrix of a system of linear equations, find the value(s) of \(k\) such that the system is consistent. (c) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. (d) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of \(k\) such that the system is consistent.

Consider the matrix \(A=\left[\begin{array}{rrr}2 & -1 & 3 \\ -4 & 2 & k \\ 4 & -2 & 6\end{array}\right]\) (a) If \(A\) is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. (b) If \(A\) is the augmented matrix of a system of linear equations, find the value(s) of \(k\) such that the system is consistent. (c) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. (d) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of \(k\) such that the system is consistent.

Describe all possible \(2 \times 2\) reduced row-echelon matrices. Support your answer with examples.

Determine conditions on \(a, b, c,\) and \(d\) such that the matrix $$\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$ will be row-equivalent to the given matrix. $$\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]$$

Find all values of \(\lambda\) (the Greek letter lambda) such that the homogeneous system of linear equations will have nontrivial solutions. $$\begin{array}{r} (\lambda-2) x+\quad y=0 \\ x+(\lambda-2) y=0 \end{array}$$