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Chapter 7: Problem 2
The _________ of an inequality is the collection of all solutions of the inequality.
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Data Analysis: Wildlife A wildlife management team studied the reproductive rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females \(x ,\) and the percent of females \(y\) that had offspring the following year were recorded. The table shows the results.$$ \begin{array} { | c | c | c | c | } \hline \text { Number, } x & { 100 } & { 120 } & { 140 } \\ \hline \text { Percent, y } & { 75 } & { 68 } & { 55 } \\\ \hline \end{array} $$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the percent of females that had offspring when there were 170 females. (d) Use the model to estimate the number of females when 40\(\%\) of the females had offspring.
Describing an Unusual Characteristic, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=3 x+4 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \leq 1} \\ {2 x+y \leq 4}\end{array} $$
Ticket Sales For a concert event, there are \(\$ 30\) reserved seat tickets and \(\$ 20\) general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to \(3000 .\) The promoter must take in at least \(\$ 75,000\) in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold.
Fitting a line to Data To find the least squares regression line \(y=a x+b\) for a set of points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) you can solve the following system for \(a\) and \(b\) $$ \left\\{\begin{array}{c}{n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\left(\sum_{i=1}^{n} y_{i}\right)} \\ {\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\left(\sum_{i=1}^{n} x_{i} y_{i}\right)}\end{array}\right. $$ In Exercises 55 and \(56,\) the sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the result. $$ \left\\{\begin{array}{c}{6 b+15 a=23.6} \\ {15 b+55 a=48.8}\end{array}\right. $$
Think About It Consider the system of equations $$\left\\{\begin{array}{l}{a x+b y=c} \\ {d x+\epsilon y=f}\end{array}\right.$$ (a) Find values for \(a, b, c, d, e,\) and \(f\) so that the system has one distinct solution. (There is more than one correct answer.) (b) Explain how to solve the system in part (a) by the method of substitution and graphically. (c) Write a brief paragraph describing any advantages of the method of substitution over the graphical method of solving a system of equations.
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